Practical estimation of a splitting parameter for a ...elie.korea.ac.kr/~cfdkim/papers/PracticalParameterTernaryCH.pdf · D.JEONGANDJ.KIM where dis an open domain in R .d D 1,2/,

Research Article

Received 19 November 2015 Published online 19 July 2016 in Wiley Online Library

(wileyonlinelibrary.com) DOI: 10.1002/mma.4093MOS subject classification: 65M06; 35K55

Practical estimation of a splitting parameterfor a spectral method for the ternaryCahn–Hilliard system with alogarithmic free energy

Darae Jeong and Junseok Kim*†

Communicated by W. Wendland

We propose a practical estimation of a splitting parameter for a spectral method for the ternary Cahn–Hilliard system witha logarithmic free energy. We use Eyre’s convex splitting scheme for the time discretization and a Fourier spectral methodfor the space variables. Given an absolute temperature, we find composition values that make the total free energy be min-imum. Then, we find the splitting parameter value that makes the two split homogeneous free energies be convex on theneighborhood of the local minimum concentrations. For general use, we also propose a sixth-order polynomial approxi-mation of the minimum concentration and derive a useful formula for the practical estimation of the splitting parameterin terms of the absolute temperature. The numerical tests are phase separation and total energy decrease with differenttemperature values. The linear stability analysis shows a good agreement between the exact and numerical solutions withan optimal value s. Various computational experiments confirm that the proposed splitting parameter estimation givesstable numerical results. Copyright © 2016 John Wiley & Sons, Ltd.

Keywords: ternary Cahn–Hilliard system; logarithmic free energy; spectral method; phase separation; optimal splitting parameter

1. Introduction

In this paper, we propose a practical estimation of a splitting parameter for a spectral method for the ternary Cahn–Hilliard (CH) systemwith a logarithmic free energy. For i D 1, 2, 3, let ci D ci.x, t/ be the i-th concentration of the ternary mixture. Then, the ternary CHsystem with a logarithmic free energy is as follows [1]:

@c1

@tD ��1, x 2 �, t > 0, (1)

�1 D � ln.c1/ � �cc1 � �2�c1 �

�

3ln.c1c2c3/, (2)

@c2

@tD ��2, (3)

�2 D � ln.c2/ � �cc2 � �2�c2 �

�

3ln.c1c2c3/, (4)

@c3

@tD ��3, (5)

�3 D � ln.c3/ � �cc3 � �2�c3 �

�

3ln.c1c2c3/, (6)

Department of Mathematics, Korea University, Seoul 136-713, Korea* Correspondence to: Junseok Kim, Department of Mathematics, Korea University, Seoul 136-713, Korea.† E-mail: [email protected]

Copyright © 2016 John Wiley & Sons, Ltd. Math. Meth. Appl. Sci. 2017, 40 1734–1745

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D. JEONG AND J. KIM

where� is an open domain in Rd .d D 1, 2/, � is the absolute temperature, �c is the critical temperature, and � is the gradient energycoefficient. Here, � , �c, and � are positive constants with � < �c. We use the periodic boundary conditions for the system. The ternaryCH system (1)–(6) can be derived from the following energy functional [2]:

E.t/ DZ�

F.c1, c2, c3/C

3XiD1

�2

2jrcij

2

!dx,

where F.c1, c2, c3/ is the free energy density of a homogeneous system with concentration c1, c2, and c3 as

F.c1, c2, c3/ D �.c1 ln c1 C c2 ln c2 C c3 ln c3/C �c.c1c2 C c2c3 C c3c1/. (7)

Generalization of the binary CH equation [3] appears first with deFontaine [4–6] and with Morral and Cahn [7]. The ternary CH systemwas applied to the prediction of microstructural evolutions in Fe–Cr–Mo ternary alloys in [8, 9]. The authors in [10] studied the effectof Mo on the microstructure evolution, and coarsening kinetics of � 0 precipitates in the Ni–Al–Mo system is studied using phase-field simulations. Using a phase-field method, the phase transformation process and morphological change of microstructure weretheoretically simulated for Fe–Al–Co ternary ordering alloy systems [11]. The microstructure simulation of spinodal decomposition wascarried out in the isothermally aged Cu-46 at.%Ni-4 at.%Fe alloys using the phase-field method [12].

The main purpose of this work is to propose an estimation of a splitting parameter for the ternary CH system with a thermodynami-cally consistent logarithmic free energy for a three-component mixture.

The contents of this paper are as follows. In Section 2, we consider a Fourier spectral method for the ternary CH system. In Section 3,various computational experiments such as phase separation, total energy decrease with different temperature values, and linear sta-bility analysis are performed, which confirm that the proposed splitting parameter estimation gives stable numerical results. Finally,conclusions are drawn in Section 4.

2. Numerical solution

For simplicity of exposition, we consider the numerical solution in one-dimensional space, � D Œa, b�. Extension to the other higher-dimensional problems are straightforward.

Let Nx be a positive even integer, h D .b� a/=Nx be the uniform mesh size,�h D fxmjxm D aCmh, 0 � m � Nxg be the set of gridpoints, and�t be the time step. Figure 1 illustrates the spatial grid. Let D c1,� D �1, D c2, and D �2. Let k

m be approximation of.xm, k�t/. Here, k means the time level. The others, k

m, �km, and k

m, are similarly defined. In this paper, we consider a Fourier spectralmethod and propose practical splitting parameter estimation for Eqs (1)–(4):

kC1m � k

m

�tD ��kC1

m , (8)

�kC1m D � ln

�k

m

�� .sC 1/�c

km C s�c

kC1m � �2�kC1

m

��

3ln��k

m km.1 �

km �

km

��,

(9)

kC1m � k

m

�tD �kC1

m , (10)

kC1m D � ln

� k

m

�� .sC 1/�c

km C s�c

kC1m � �2� kC1

m

��

3ln��k

m km.1 �

km �

km

��,

(11)

where s is a positive number. Here, we only solve the equations with and because the third concentration can be defined as1 � � .

2.1. Fourier spectral method

For the given data fk1 ,k

2 , � � � ,kNxg, the discrete Fourier transform is defined as

Okp D

NxXmD1

kme�i�pxm ,

Figure 1. Uniform grid fx0, x1, � � � , xNx gwith h D xmC1 � xm for m D 0, � � � , Nx � 1.


17

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D. JEONG AND J. KIM

Figure 2. (a) Free energy surface F.c1, c2, c3/, (b) its contour plot, and (c) free energy F.�,�, 1� 2�/ and �min on line l.

where �p D 2p�=.b � a/. The inverse discrete Fourier transform is

km D

1

Nx

Nx=2�1XpD�Nx=2

Okp ei�pxm . (12)

Let us assume that

.x, k�t/ D1

Nx

Nx=2�1XpD�Nx=2

Okp ei�px . (13)

Then, the second-order and fourth-order partial derivatives are given as

�.x, k�t/ D �1

Nx

Nx=2�1XpD�Nx=2

�2pOk

p ei�px , (14)

�2.x, k�t/ D1

Nx

Nx=2�1XpD�Nx=2

�4pOk

p ei�px . (15)

By substituting Eqs (13)–(15) into Eqs (8) and (9), we have

OkC1p � Ok

p

�tD ��2

pOf kp � �

2p s�c O

kC1p � �4

p�2 OkC1

p .

Here, Of kp D

PNxmD1f k

me�i�pxm , where f km D � ln.k

m/ � .sC 1/�ckm � � ln

��k

m km.1 �

km �

km

��=3. Then, we have

OkC1p D

�Ok

p=�t � �2pOf kp

�=�1=�tC �2

p s�c C �4p�

2�

. (16)

Finally, using Eqs (12) and (16), we obtain .kC 1/-th time step solution, kC1m for m D 1, � � � , Nx . Similarly, we can obtain kC1

m .

2.2. Optimal splitting factor s�

By Eyre’s theorem [13, 14], if we can split the free energy appropriately into contractive and expansive parts and then treat the con-tractive part implicitly and the expansive part explicitly, the numerical splitting algorithm is unconditionally gradient stable. In theright-hand side of Eq. (9), � ln.k/ � .s C 1/�c

k C s�ckC1 can be rewritten as F0c.

kC1/ � F0e.k/, where Fc./ D 0.5s�c

2 is a con-tractive term and Fe./ D 0.5.sC 1/�c

2 � � ln./C � is an expansive term. Note that both functions Fc./ and Fe./ are strictlyconvex in the region satisfying s > 0, �c > 0, and�

2 .0, 1/ : F00e ./ D .sC 1/�c ��

> 0

. (17)

To find the optimal splitting factor, we use the minimum point min as the minimum point at which F.,, 1 � 2/ achieves aminimum; see Eq. (7) for the definition of F. Figure 2 represents the free energy surface F.c1, c2, c3/ and its contour plot. As shown inFigure 3, the free energy has three local minima. By Cahn–Hilliard dynamics, each ci moves the position of the local minima to minimizethe energy. To minimize the local discretization error [13], we need to take the value of s as small as possible. At the same time, we wantto keep the strictly convexity of Fe./, that is, Eq. (17). Therefore, in the proposed method, we compute the minimum point min, andbased on the value, we define the splitting parameter s.


17

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D. JEONG AND J. KIM

Figure 3. Minimum points �min (circled symbol) with respect to � and a sixth-order best-fitted polynomial (solid line) when �c D 1.

Figure 4. Practically stable region of s over � when �c D 1. The dashed line denotes optimal splitting factor s� against � .

Let g./ D F.,, 1 � 2/, then

g0.min/ D 2� ln

min

1 � 2min

�C �c.2 � 6min/ D 0. (18)

We numerically solve Eq. (18) by using fzero command in MATLAB [15]. Figure 3 shows the minimum points min (circled symbol)with respect to � and a sixth-order best-fitted polynomial (solid line) when �c D 1. Here, using polyfit command in MATLAB, we findthe polynomial of order 6 that fits best to the given data points .� ,min/ as

p.�/ D 9420.12298 �6 � 13037.1741 �5 C 7460.65877 �4

� 2249.07797 �3 C 377.41886 �2 � 33.41158 � C 1.21763,(19)

for 0.15 � � � 0.35. For the reader unfamiliar with MATLAB, we provide the code in the Appendix.Now, by the condition (17) and approximated polynomial (19), we obtain the condition of splitting factor s satisfying convexity of

Fe./ as

s >�

min�c� 1 �

�

p.�/�c� 1. (20)

In Figure 4, practically stable region is shown in the phase plane of temperature � and splitting parameter s using inequality Eq. (20).We define an optimal splitting factor s� as s� D 1.1�=.�cmin/� 1, where 1.1 is a safety factor. Therefore, for a given temperature � , wecan automatically decide the splitting factor s. For example, when �c D 1, the optimal splitting factor s� is

s� D 1.1�=p.�/ � 1.

In Figure 4, we can also see �=.�cmin/ � 1 and 1.1�=.�cmin/ � 1 with solid and dashed lines, respectively. We note that the authorin [16] suggested choosing optimal values of splitting parameter as a future research.

Note that our strategy for the optimal splitting parameter is based on the assumption � min. In fact, there is no maximal princi-ple in the ternary CH equation. However, the minimum values of the phase stay in the neighborhood of min during evolution. Eventhough we derive the convex condition with the assumption � min, effective assumption is that � min=1.1 because of thesafety factor.


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D. JEONG AND J. KIM

3. Numerical experiments

In this section, we present various computational experiments to demonstrate that the proposed splitting parameter estimation givesstable numerical results. The numerical tests are spinodal decomposition and total energy decrease with different temperature values.

3.1. One-dimensional space

First, we study phase separation dynamics on one-dimensional space � D .0, 2/ with different � . In this numerical test, we use Nx D

200, h D 0.01,�t D 0.1, and � D h. The initial conditions are set to

.x, 0/ D 0.25C 0.1rand.x/, .x, 0/ D 0.25C 0.1rand.x/, (21)

where rand.x/ is a random number generated uniformly in interval Œ�1, 1�. Figure 5 represents the temporal evolution of numericalsolutions in the ternary system with � D 0.15, 0.225, and 0.3. The times are shown at the bottom of each figure. In early times, we canobserve two phases; one of them is dominated by 1 � � , and the other phase is dominated by � . Then, we can see threedominating phases at later times. Moreover, as the absolute temperature value � increases, maximum concentration value decreases.With increasing absolute temperature values, we have decreasing maximum concentration values.

We define a discrete energy functional as

Eh.k , k/ D hNxX

mD1

��k

m lnkm C

km ln k

m C�1 � k

m � km

�ln�1 � k

m � km

��

C�c

�k

m km C

�k

m C km

� �1 � k

m � km

��C�2

2

�ˇ̌rk

m

ˇ̌2Cˇ̌r k

m

ˇ̌2Cˇ̌rk

m Cr km

ˇ̌2��,

where

rkm D

1

Nx

Nx=2�1XpD�Nx=2

i�p Okp ei�pxm .

Figure 5. Numerical solutions of the ternary system with (a) � D 0.15, (b) � D 0.225, and (c) � D 0.3. The times are shown at the bottom of each figure.


17

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D. JEONG AND J. KIM

Figure 6. Scaled discrete energies E�.�k , k/ of the numerical solutions on a ternary system with � D 0.15, 0.225, and 0.3. Here, we use the initialconditions (21).

By Eq. (22), we calculate the temporal evolutions of the discrete energy Eh.k , k/with � D 0.15, 0.225, and 0.3. For comparison, wedefine the scaled energy as

E�.k , k/ DEh.k , k/ � Eh.Nt , Nt /

Eh.0, 0/ � Eh.Nt , Nt /. (22)

These results are shown in Figure 6. In all cases, the scaled energy E�.k , k/ decreases as time goes on.Next, we consider three equal concentrations as the initial condition:

.x, 0/ D 1=3C 0.1rand.x/, .x, 0/ D 1=3C 0.1rand.x/.

The other parameters are set to the same values used in the previous numerical test. Figure 7 represents the temporal evolutions of thenumerical solutions of the ternary system with � D 0.15, 0.225, and 0.3. The times are shown at the bottom of each figure. We can seethree phases in the early stages of phase separation. Figure 8 shows the temporal evolutions of the scaled discrete energy E�.k , k/

with � D 0.15, 0.225, and 0.3. They are decreasing as we expected.

3.2. Two-dimensional space

In this section, we present the computational results of spinodal decomposition in ternary systems on two-dimensional space. Thesimulations are performed on domain � D Œ0, 2� � Œ0, 2� using Nx � Ny D 200 � 200 grid points with h D 0.01, �t D 0.1, and � D h.Figure 9 represents the temporal evolution of the numerical solutions in the ternary system with the initial conditions:

.x, y, 0/ D 0.25C 0.05rand.x, y/, .x, y, 0/ D 0.25C 0.05rand.x, y/,

where rand.x, y/ is the random value in Œ�1, 1�. To verify the stability of the numerical solution with respect to the given temperature,we perform the numerical simulations with � D 0.15, 0.225, and 0.3. By using the proposed optimal splitting factor s�, we obtain thestable spinodal decomposition dynamics. Note that we have binary-like phase separation in early times.

In two-dimensional space, we also define a discrete energy functional as follows:

Eh.k , k/ D h2

NyXnD1

NxXmD1

��k

mn lnkmn C

kmn ln k

mn C�1 � k

mn � kmn

�ln�1 � k

mn � kmn

��

C�c

�k

mn kmn C

�k

mn C kmn

� �1 � k

mn � kmn

��C�2

2

�ˇ̌rk

mn

ˇ̌2Cˇ̌r k

mn

ˇ̌2Cˇ̌rk

mn Cr kmn

ˇ̌2��,

where jrkmnj

2 D�rx

kmn

�2C�ry

kmn

�2. Here,

rxkmn D

1

Nx Ny

Ny=2�1XqD�Ny=2

Nx=2�1XpD�Nx=2

i�p Okpqei.�pxmC�qyn/,

rykmn D

1

Nx Ny

Ny=2�1XqD�Ny=2

Nx=2�1XpD�Nx=2

i q Okpqei.�pxmC�qyn/.

Figure 10 shows the scaled energy (22) with � D 0.15, 0.0225, and 0.3. In all cases, we see that the scaled energy E� is decreasing astime goes on.

We present the second simulation on two-dimensional space with the initial condition as

.x, y, 0/ D 1=3C 0.05rand.x, y/, .x, y, 0/ D 1=3C 0.05rand.x, y/.


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D. JEONG AND J. KIM

Figure 7. Numerical solution of a ternary system with (a) � D 0.15, (b) � D 0.225, and (c) � D 0.3. The times are shown at the bottom of each figure.

Figure 8. Scaled discrete energies E�.�k , k/ of the numerical solutions on the ternary system with � D 0.15, 0.225, and 0.3.

As shown in Figure 11, we can see the phase separation of three phases from the early stages.Figure 12 shows the scaled discrete energy of numerical solution in a ternary system. As we expected, the energies decreases as time

goes on.

3.3. Effect of splitting parameter s

In this section, we show the effect of splitting parameter s. To show the effect, we perform a linear stability analysis. With the meanconcentration as m D .m, m, 1 � 2m/, we will find a solution having the following form

.x, t/ .x, t/

�D

mm

�C

1XkD1

˛k.t/ cos.k�x/ˇk.t/ cos.k�x/

�, (23)


17

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D. JEONG AND J. KIM

Figure 9. Numerical solution of a ternary system at each time with (a) � D 0.15, (b) � D 0.225, and (c) � D 0.3. The times are shown at the bottom of eachfigure. [Colour figure can be viewed at wileyonlinelibrary.com]

Figure 10. Scaled discrete energy of numerical solution on a ternary system with � D 0.15, � D 0.225, and � D 0.3.

where j˛k.t/j, jˇk.t/j � 1 [17]. After linearizing the nonlinear terms in Eqs (1)–(4) about .m, m, 1 � 2m/, we have

@

@tD ��, (24)

� D � ln.m/C�. �m/

m� �c � �

2�

��

3

lnŒm2.1 � 2m/�C

.1 � 3m/. �m/

m.1 � 2m/C.1 � 3m/. �m/

m.1 � 2m/

�,

(25)


17

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D. JEONG AND J. KIM

Figure 11. Numerical solution of a ternary system at each time with � D 0.15, 0.225, and 0.3. The times are shown at the bottom of each figure. [Colour figurecan be viewed at wileyonlinelibrary.com]

Figure 12. Scaled discrete energy of numerical solution on a ternary system with � D 0.15, 0.225, and 0.3.

@

@tD �, (26)

D � ln.m/C�. �m/

m� �c � �

2�

��

3

lnŒm2.1 � 2m/�C

.1 � 3m/. �m/

m.1 � 2m/C.1 � 3m/. �m/

m.1 � 2m/

�.

(27)


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D. JEONG AND J. KIM

Figure 13. Temporal evolution of the numerical amplitude ˛2.t/with different splitting factor s and its corresponding result by linear stability analysis.

Figure 14. Phase transformation of (a) �, (b) , and (c) 1� � � in metastable region.

By substituting Eq. (23) into Eqs (24)–(27), we obtain˛0k.t/ˇ0k.t/

�D A

˛k.t/ˇk.t/

�, where A D

a bb a

�, (28)

and a D �.k�/2

��.2 � 3m/

3m.1 � 2m/� �c C �

2.k�/2

, b D

.k�/2�.1 � 3m/

3m.1 � 2m/.

The eigenvalues of A are �1 D a � b and �2 D aC b. The solution to the system of the ordinary differential equations (28) is given by˛k.t/ˇk.t/

�D

e�1t

2

�˛k.0/C ˇk.0/˛k.0/ � ˇk.0/

�C

e�2t

2

˛k.0/C ˇk.0/˛k.0/C ˇk.0/

�.

For the numerical test, we take the initial condition as

.x, 0/ D mC 0.01 cos.k�x/, .x, 0/ D mC 0.01 cos.k�x/ on� D .0, 1/.

The other parameters are m D 0.25, k D 2, h D 1=1000, � D h, � D 0.3, �c D 1.0, and its corresponding optimal splitting factors� � 4.9972. The numerical simulations are run up to T D n�t D 0.01 with �t D 0.0001. Figure 13 shows the temporal evolution ofthe numerical amplitudes ˛k.t/ and its corresponding linear stability analysis results. When s D s�, the results are in good agreementin a linear regime. Also, through this test, we can see the dependence of the numerical solutions on the splitting factor s. As s is larger,


17

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D. JEONG AND J. KIM

the numerical solution has smaller growth rate than the linear analysis. In other words, the larger splitting parameter s makes slowtime evolution for numerical solution by generating a larger local truncation error. Therefore, we can confirm that s� is an optimalsplitting parameter.

3.4. Phase transformation in metastable region

As the last test, we consider phase transformation in metastable region, that is, .x, 0/ < min. Let the initial condition be given as.x, 0/ D 0.03C 0.01 cos.20�x/ and .x, 0/ D 0.03C 0.01 cos.10�x/ on the one-dimensional space� D Œ0, 1�. The other parametersare Nx D 1000, h D 1.0e-3,�t D 1.0e-6, T D 0.002, � D h, � D 0.3, and �c D 1. Figure 14(a)–(c) represents the temporal evolution for, , and 1 � � , respectively. Because the initial condition is set in metastable region, we obtain the stable solution against initialperturbations as time goes on. Therefore, we can check that our algorithm works fine in the region a little away from min.

4. Conclusions

In this paper, we proposed a practical estimation of a splitting parameter for a spectral method for the ternary CH system with alogarithmic free energy. By Eyre’s convex splitting scheme, we derived a useful formula for the practical estimation of the splittingparameter in terms of the absolute temperature. In the previous research, finding an optimal splitting parameter has been carriedout by trial and error. However, in this study, we can choose automatically an optimal splitting parameter by the derived formula. Thenumerical tests such as phase separation and total energy that decrease with different temperature values demonstrated that theproposed splitting parameter estimation gives stable numerical results. The linear stability test showed that the numerical solutionwith optimal value s has a good agreement with exact solution.

Appendix A. MATLAB code

The below MATLAB script is composed of 19 lines that are divided into two parts: (i) root-finding of equation (18) with fzero commandand (ii) approximating a sixth-order best-fitted polynomial of the given data with polyfit command in MATLAB [15].

clear all; theta_c=1; n=41; theta=linspace(0.15,0.35,n);

gprime=@(x,t) 2*t*log(x/(1-2*x))+theta_c*(2-6*x)

for k=1:n

fuc=@(x) gprime(x,theta(k));

flag=1; iniguess=0.1;

while flag==1

phimin(k)=fzero(fuc,iniguess);

if (phimin(k)>0 || phimin(k)<0)

flag=0;

else

iniguess=iniguess/10;

end

end

end

p = polyfit(theta,phimin,6);

x1 = linspace(theta(1),theta(end),200); y1 = polyval(p,x1);

plot(theta,phimin,’ko’); hold on

plot(x1,y1,’k-’); axis([theta(1) theta(n) 0 max(phimin)])

legend(’data’,’polynomial fit’,’Location’,’northwest)’

Acknowledgements

The first author (D. Jeong) was supported by a Korea University grant. The corresponding author (J.S. Kim) was supported by theNational Research Foundation of Korea (NRF) grant funded by the Korea government (MSIP) (NRF-2014R1A2A2A01003683). The authorsgreatly appreciate the reviewers for their constructive comments and suggestions, which have improved the quality of this paper.

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