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INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING Int. J. Numer. Meth. Engng (2016) Published online in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/nme.5372

Numerical approximations for a phase field dendritic crystal growth model based on the invariant energy quadratization approach Jia Zhao1,2 , Qi Wang1,3,4 and Xiaofeng Yang1,*,† 1 Department

of Mathematics, University of South Carolina, Columbia, SC 29208, USA of Mathematics,University of North Carolina, Chapel Hill, NC, 27599, USA 3 School of Mathematical Sciences, Nankai University, Tianjin, China 300071 4 Beijing Computational Science Research Center, Beijing, China 100193

2 Department

SUMMARY We present two accurate and efficient numerical schemes for a phase field dendritic crystal growth model, which is derived from the variation of a free-energy functional, consisting of a temperature dependent bulk potential and a conformational entropy with a gradient-dependent anisotropic coefficient. We introduce a novel Invariant Energy Quadratization approach to transform the free-energy functional into a quadratic form by introducing new variables to substitute the nonlinear transformations. Based on the reformulated equivalent governing system, we develop a first and a second order semi-discretized scheme in time for the system, in which all nonlinear terms are treated semi-explicitly. The resulting semi-discretized equations consist of a linear elliptic equation system at each time step, where the coefficient matrix operator is positive definite and thus, the semi-discrete system can be solved efficiently. We further prove that the proposed schemes are unconditionally energy stable. Convergence test together with 2D and 3D numerical simulations for dendritic crystal growth are presented after the semi-discrete schemes are fully discretized in space using the finite difference method to demonstrate the stability and the accuracy of the proposed schemes. Copyright © 2016 John Wiley & Sons, Ltd. Received 29 April 2016; Revised 25 August 2016; Accepted 31 August 2016 KEY WORDS:

Phase-field models; Dendritic Crystal Growth; Unconditional Energy Stability; Linear Elliptic Equations; Second Order; Invariant Energy Quadratization.

1. INTRODUCTION A dendrite is a crystal with a typical multi-branching tree-like structure. Dendritic crystallization usually forms natural fractal microstructure, which is very common and observable in natural phenomena like snowflake formation or frost patterns on a window. Microstructures formed during solidification (freezing) of a molten material play an enormous role in properties of the final solid material. Specifically, during the solidification of an alloy, micro-segregation patterns that result during dendritic solidification of an alloy are of substantial interests to material processing communities. Dendritic crystals stem from growth instabilities that occur when the growth rate is limited by the rate of diffusion of solute atoms to the interface. When the liquid (or the molten material) is supercooled below the freezing point of the solid, a spherical solid nucleus grows in the under-cooled melt initially. As the sphere grows, the spherical morphology becomes unstable and its shape is perturbed. Along some preferred growth directions, the solid shape begins to express some protuberance accompanied by steeper concentration gradients at its tip. The growth direction may be due *Correspondence to: Xiaofeng Yang, Department of Mathematics, University of South Carolina, Columbia, SC 29208, USA. † E-mail: [email protected] Copyright © 2016 John Wiley & Sons, Ltd.

J. ZHAO, Q. WANG AND X. YANG

to anisotropy in the surface energy of the solid–liquid interface, and/or to the ease of attachment of atoms to the interface on different crystallographic planes. If the anisotropy is large enough, the dendrite then exhibits a sharper and sharper tip as it grows. To understand the process of dendritic crystal growth, many theoretical models have been developed and numerical simulations based on these models were carried out (cf. [1–12] and the references therein) to study transport of heat and solute in the vicinity of the dendrite tip and the factors that control the stability of the shape of the dendrite tip, and to determine the tip radius and initial secondary spacing. In a pioneering work [2], the diffuse interface approach, also known as the phase field method, was first applied in the solidification problem that accounts for the Gibbs–Thomson equation at the solid–liquid interface for solidification problems. Subsequently, the phase field dendritic crystal growth model (PF–DCG) has been improved quite a bit in the past two decades. In [4, 7, 10, 12], based on the time dependent Ginzburg–Landau mesoscopic simulation method, the authors developed various phase-field models that exhibited great versatility to study dynamics of atomic-scale dendritic crystal growth dependent on temperature and isotropic/anisotropic diffusive time scales. The essential idea of the PF–DCG model is that an order parameter (phase field variable) is adopted to define the physical state (liquid or solid) of the system at each point, and a free-energy functional is devised by incorporating a specific form of the conformational entropy with anisotropic spatial gradients. An advantage of the phase field approach is that the governing system of equations in the model can be naturally derived from a variational principle. As a result, the system satisfies energy dissipation laws, which justify its thermodynamic consistency and lead to a mathematically well-posed model. The existence of the energy law can then serve as the guidance for the design of energy stable numerical schemes. Apparently, it is physically and mathematically desirable to design numerical schemes that respect the energy dissipation law of the original PDEs at the discrete level. On the one hand, preserving the energy law is critical for the numerical schemes to capture the correct long time dynamics of the system. On the other hand, the unconditional stability of the energy-law preserving schemes provides flexibility for dealing with the stiffness issue (associated with the thin interfacial width) in phase-filed models. Although many numerical works had been devoted to solve the PF–DCG model effectively, because of the complexity of the nonlinear terms in the model, there has not been a successful attempt to develop efficient and accurate schemes with a rigorously proven unconditionally energy stability property. Most of the available schemes are either nonlinear which need some efficient iterative solvers, and/or does not preserve energy stability at all (cf. [1, 6, 13–17] and the references therein). Therefore, the main focus of this paper is to develop efficient numerical schemes to solve the PF–DCG system that can preserve energy stability at the discrete level. To this end, we adopt a novel approach, that we term it the Invariant Energy Quadratization (IEQ) approach, where some nonlinear transformations are introduced to enforce the free-energy density as an invariant, quadratic functionals in terms of new, auxiliary variables. The IEQ method has been successfully applied in the context of other models in the authors’ other work [18–22], while the application to the PF-DCG model provides new challenges because of the nonlinearities in various terms, in particular, the anisotropic coefficient in the gradient entropy, the fourth order and fifth order nonlinear polynomial potentials. The key point of IEQ method is that we reformulate the governing system of equations using the new variables to an equivalent system. Thereafter, we discretize the governing system of equations in the new variables by treating all nonlinear terms semi-explicitly, which in turn produces a system of linear elliptic equation system. We show that the linear operator is symmetric and positive definite so that the equation system can be solved using fast elliptic solvers efficiently. Using this new strategy, we develop a series of linear numerical schemes which preserve the energy dissipation law at arbitrary time step sizes (a desired property known as unconditional energy stability), while avoiding employing artificial stabilizers (cf. [23–27]) or conducting convex splitting iterations (cf. [28–32]). In summary, the new numerical schemes that we develop in this study all possess the following properties: (1) the schemes are accurate (up to second order in time); (2) they are unconditionally stable (energy dissipation law holds for arbitrary time steps); and (3) they are efficient and easy to implement (ones only need to solve some linear elliptic equations at each time step). To the best of our knowledge, the proposed schemes are the first such schemes for solving the PF–DCG system that can have all these desired properties. Copyright © 2016 John Wiley & Sons, Ltd.

Int. J. Numer. Meth. Engng (2016) DOI: 10.1002/nme

NUMERICAL APPROXIMATIONS FOR A DENDRITIC CRYSTAL GROWTH MODEL

The rest of the paper is organized as follows. In Section 2, we give a brief introduction of the PF– DCG model. Then, we reformulate it to the equivalent system via the IEQ approach, and present its energy dissipation law. In Section 3, we propose the numerical schemes and prove their unconditional energy stability and solvability in the time discrete case. In Section 4, we present the convergence test result and various numerical simulations of dendritic crystal growth in 2D and 3D to demonstrate the accuracy and efficiency of the new schemes. Finally, some concluding remarks are presented in Section 5. 2. THE MODEL SYSTEM 2.1. The phase field dendritic crystal growth model We first briefly describe the phase field model for dendritic crystal growth [4, 7, 10, 12]. In a confined domain ˝ 2 Rd , .d D 2; 3/, a phase variable (or a labeling function) !.x; t / (x 2 ˝, t is the time) is introduced to label the liquid and solid phase, respectively: ² !1 fluid, !.x; t / D (2.1) 1 solid; with a smooth but steep transitional layer controlled by parameter ". We consider the following free energy: # Z " 1 2 E.!; T / D (2.2) " j".r!/r!j2 C F .!/ C #"T G.!/ d x; 2 ˝ where T .x; t / is the temperature, a prescribed field in this study, F .!/ D 14 .1 ! ! 2 /2 is the double well potential, and G.!/ D 15 ! 5 ! 23 ! 3 C ! is a fifth order polynomial function accounting for the generation of latent heat. We note that, in some papers (cf. [4]), the function G.!/ takes on an alternative form (e.g., G.!/ D ! ! 13 ! 3 ). By assuming the conformational entropy coefficient, ".r!/ depends on the direction of the outer r! ) , the anisotropy is effectively introduced into the normal vector n at the interface (n " ! jr!j system. In 2D, the anisotropy of conformational entropy is usually given by " D 1 C "4 cos.m$/;

(2.3)

where m is the number of folds of anisotropy, "4 is a parameter for the anisotropy strength, and @ ! $ defined as tan.$.!// D @yx ! in 2D (cf. [7]). One can easily derive when m D 4, the fourfold symmetric model-type anisotropy is given by the following: in 2D (cf. [3, 4, 7]), 4"4 !x4 C !y4 ".r!/ D .1 ! 3"4 / 1 C 1 ! 3"4 jr!j4

!

(2.4)

and an analogy in 3D (cf. [6]) is given as 4"4 !x4 C !y4 C !´4 ".r!/ D .1 ! 3"4 / 1 C 1 ! 3"4 jr!j4

!

:

(2.5)

Adopting a relaxation dynamics for the dendritic crystal growth, one obtains the governing dynamical equation in the PF–DCG model using the variational approach as follows: ıE.!; T / ı! $ % $ % D "2 r # " 2 .r!/r! C "2 r # jr!j2 ".r!/H .!/ ! f .!/ ! #"T g.!/;

% .$.!//!t D !

Copyright © 2016 John Wiley & Sons, Ltd.

(2.6)



where 8 $ % 0 2 ˆ < f .!/ D F 0 .!/ D ! 21 ! !2 ; g.!/ D G .!/ D .!& ! 1/ ; ' ˆ : H .!/ D ı".r!/ D ı".r!/ ; ı".r!/ ; ı".r!/ ; ır!

ı!x

ı!y

(2.7)

ı!´

and % .$.!// > 0 is the mobility coefficient chosen either as a constant (cf. [4, 12]) or as a function of the angle $. In 2D, we have H .!/ D 4"4 whereas in 3D,

% $ %% 4 $ $ 2 2 !x !x !y ! !y4 ; !y !x2 !y2 ! !x4 ; 6 jr!j

% 4 $ $ 2 2 !x !x !y C !x2 !´2 ! !y4 ! !´4 ; 6 jr!j $ 2 2 % $ %% !y !y !´ C !x2 !y2 ! !x4 ! !´4 ; !´ !x2 !´2 C !y2 !´2 ! !x4 ! !y4 :

H .!/ D4"4

(2.8)

(2.9)

The equation for temperature field T is derived from the conservation of enthalpy as follows: ut D Du &T C K!t ;

(2.10)

where Du is the diffusion coefficient, K is a dimensionless latent heat coefficient that is inversely proportional to the strength of cooling. For system (2.6)–(2.10), we adopt the periodic boundary condition to remove all complexities associated with the boundary integrals in this study. We remark that the boundary conditions can also be the no-flux boundary condition of @n !j@˝ D 0. Remark 1 We emphasize (2.6) is the only equation that is derived by the variational approach from the freeenergy functional (2.2). Therefore, one can only develop the energy stable schemes to solve (2.6) instead of the coupled system (2.6)–(2.10). For simplicity, we assume the temperature field T .x/ is a positive function independent of time when we show the energy stability of the proposed schemes. For (2.6), by taking the L2 inner product of (2.6) with !t , we can easily obtain the energy dissipation law as follows, p d E.!; T / D !k % .$.!// !t k2 $ 0: dt

(2.11)

2.2. The Reformulated equivalent model We rewrite free energy (2.2) by adding two zero terms, 12 "2 .A ! A/ and #"T 16 .! 6 ! ! 6 /, and then express the free-energy density as a quadratic form as follows: Z " '2 1 %2 1$ 2 1 2 &p E.!; u/ D j".r!/r!j2 C A ! "2 A C " ! !1 2 2 4 ˝ !2 !2 1 r r (2.12) 2 1 6 1 5 1 6 A d x: C#"T ! #"T ! C ! CB ! ! ! C !3 C B 6 5 6 3 where A and B are two positive constants to ensure the positivity of j".r!/r!j2 CA, 16 ! 6 C 15 ! 5 C B and 16 ! 6 ! ! C 23 ! 3 C B, respectively. For example, A D 1; B D 1 can meet such a requirement.




We emphasize that the free energy is invariant because we simply add zero terms therein. We then define four scalar auxiliary functions U.!/, V .!/, W .!/ and Z.!/ as follows: p 8 U.!/ D j".r!/r!j2 C A; ˆ ˆ ˆ 2 ˆ ! 1; < V .!/ D !q (2.13) 1 6 W .!/ D 6 ! C 15 ! 5 C B; ˆ ˆ q ˆ ˆ : Z.!/ D 1 ! 6 ! ! C 2 ! 3 C B: 6

3

Thus, the free energy E is now quadratized in the four new variables as follows: E.T; U; V; W; Z/ D

Z " ˝

# "2 A 1 2 2 1 2 2 2 d x: " U C V C #"T W ! #"T Z ! 2 4 2

(2.14)

We then rewrite the PF–DCG equation in the following equivalent form using the new variables: % .$.!//!t D "2 r # .UR/ ! V ! ! #"P .!/W C #"uQ.!/Z;

(2.15)

"2 Ut D "2 R # r!t ;

(2.16)

1 Vt D !!t ; 2

(2.17)

2#"T Wt D #"TP .!/!t ;

(2.18)

! 2#"T Zt D !#"TQ.!/!t ;

(2.19)

where 8 2 2 ".r!/H .!/ rˇ ; R.!/ D " .r!/r!Cjr!j ˆ ˇ2 ˆ ˆ ˇ ˆ ".r!/r! ˇ CA ˆ < 5 4 ; P .!/ D q 1 ! C! 1 6 5 ˆ 6 ! C 5 ! CB ˆ ˆ 5 2 !1 ˆ ˆ : Q.!/ D q 1 ! C2! : 2 6!

6C

3!

3 !!CB

The initial conditions are given by 8 !.t D 0/ D !p0 ; ˆ ˆ ˆ ˆ U.t D 0/ D j".r!0 /r!0 j2 C A; ˆ ˆ < 2 V .t D 0/ D !q 0 ! 1; ˆ ˆ W .t D 0/ D 16 !06 C 15 !05 C B; ˆ ˆ q ˆ ˆ : Z.t D 0/ D 1 ! 6 ! ! C 2 ! 3 C B: 6 0

(2.20)

0

(2.21)

3 0

The boundary conditions are periodic or no-flux boundary conditions @n ! nC1 j˝ D 0 as we alluded to before. Remark 2 We consider the nonlinear functionals in the free energy (2.12) as four quadratic functionals by applying appropriate substitutions if needed. Therefore, after simple substitutions using new variables U; U; W; Z defined in (2.13), the energy is transformed to an equivalent quadratic form. Therefore, we call it ‘Invariant Energy Quadratization’ approach (see also the authors other work [18–22, 33]). Copyright © 2016 John Wiley & Sons, Ltd.



We emphasize that the positive constant A and B must be introduced in the new variables to avoid the appearance of singularities in R.!/, P .!/, and Q.!/ because their denominators might equal to zero without them. The energy dissipation law given in (2.11) retains. In fact, we can derive it by taking the L2 inner product of (2.15) with !t , of (2.16) with U , of (2.17) with V , of (2.18) with W , of (2.19) with Z, and then perform integration by parts. Summing over all equalities, we arrive at (2.11) in new variables: p d E.T; U; V; W; Z/ D !k % .$.!//!t k2 $ 0: dt

(2.22)

In the following, we devise two semi-discretized numerical schemes in time for the equivalent system (2.15)–(2.19), which satisfy a discrete analog of (2.22). The proof of unconditional energy stability of the schemes follow exactly the procedure outlined earlier for deriving energy law (2.22). We denote the RL2 inner product of any two spatial functions f1 .x/ and f2 .x/ by 2 .f p1 .x/; f2 .x// D ˝ f1 .x/f2 .x/d x , and the L norm of the function f1 .x/ by kf1 k D .f1 ; f1 /. 3. NUMERICAL SCHEMES

We now construct semi-discrete numerical schemes in time for solving the equation system consisting of (2.15)–(2.19) and establish their energy stabilities. In this paper, we only consider the semi-discrete case in time. It can be shown that energy stability of the semi-discrete schemes is also valid when fully discretized, for appropriate finite element or spectral spatial discretizations because all proposed schemes are linear, and the trial functions are in the space as the unknowns. To this end, the main challenge for the model system (2.6) (or the equivalent system (2.15)–(2.19)) is that we need find suitable approaches to discretize four complicated nonlinear terms, including the cubic polynomial term f .!/, the fourth order polynomial term g.!/, and the other two nonlinear terms associated with the gradient entropy, that is, " 2 .r!/r! and jr!j2 ".r!/H .!/. The discretization of the cubic polynomial term f .!/ have been well studied in previous works (cf. [24, 34–37]). Briefly, there are mainly two commonly used techniques to discretize f in order to preserve unconditional energy stability. The first technique is the well-known convex splitting approach [29], where f is decomposed into a difference of a convex function and a concave function; the convex part of the potential is treated implicitly while the concave part is treated explicitly. The convex splitting approach can yield an energy stable scheme at the expense that it produces a nonlinear scheme in most cases. As the result, the implementation is complicated and the computational cost is high. The second technique is the so-called stabilized approach [23–25], where the term from the nonlinear potential is simply treated explicitly and some linear stabilizing terms are added to enforce stability. Such linear schemes are easy to implement; however, the magnitude of the stabilizing term depends on the upper bound of the second order derivative of the nonlinear potential. We notice that the double well potential in the free-energy density does not have any upper bounds. To remedy the situation, one has to make some reasonable revisions to the potential, for example, modifying potential to be of the quadratic order far away [23–25]. Such a method is particularly reliable for the models with the maximum principle. If the maximum principle does not exist, the revision to the nonlinear potentials may lead to spurious solutions. Moreover, its second order scheme only preserves conditional energy stability (cf. [24]), where the time step is limited by the interfacial thickness. Nonetheless, neither method is an optimal choice for PF–DCG model (2.6). First, it is uncertain whether the PDE solution satisfies the maximum principle. Second, it is not clear how the anisotropic conformational entropy (" 2 .r!/jr!j2 ) in the free-energy density can be split into a convex and concave pair. In this study, our objective is to develop numerical schemes that could possess the advantages of both the convex splitting approach and the stabilized approach, but bypass their difficulties mentioned earlier. More precisely, we expect that the schemes are efficient (linear system), Copyright © 2016 John Wiley & Sons, Ltd.



unconditionally energy stable (rigorously proved), and accurate (ready for second order or even higher order in time). After we coin the new IEQ approach, as a result, whether or not the governing system of PDEs admits the maximum principle or the free-energy density yields a convexity/concavity splitting is immaterial. 3.1. First-order scheme We present the first order time stepping scheme to solve system (2.15)–(2.19). Let ıt > 0 denote the time step size and set t n D n ıt for 0 $ n $ N with T D N ıt . Assume that temperature field T .x/ is a positive function independent of time and ! n , U n , V n , W n , Zn are already calculated, we compute ! nC1 , U nC1 , V nC1 , W nC1 , Z nC1 from the following time-stepping scheme with periodic boundary conditions: Step 1: %n

! nC1 ! ! n D "2 r # .U nC1 Rn / ! V nC1 ! n ! #"TP n W nC1 C #"TQn Zn ; ıt

(3.1)

U nC1 ! U n r! nC1 ! r! n D "2 Rn # ; ıt ıt

(3.2)

"2

! nC1 ! ! n 1 V nC1 ! V n D !n ; 2 ıt ıt

2#"T

(3.3)

! nC1 ! ! n W nC1 ! W n D #"TP n ; ıt ıt

(3.4)

where % n D % .$.! n // ; Rn D R.! n /; " n D ".r! n /; P n D P .! n /; Qn D Q.! n /: Step 2: ! 2#"u

! nC1 ! ! n ZnC1 ! Zn D !#"TQn : ıt ıt

(3.5)

Remark 3 In step 1, we solve a linear coupled system for !; U; V , and W . Because step 2 is a simple linear equation, one obtain ZnC1 readily after ! nC1 is calculated. Step 1 ((3.1)–(3.4)) and Step 2 ((3.5)) are totally decoupled by treating Q.!/ and Z.!/ explicitly. As it is shown below, linear system (3.1)– (3.4) in step 1 is positive definite as well. Therefore, one can apply any Krylov subspace methods with mass lumping as pre-conditioners to solve such a system efficiently. We note that ! nC1 , U nC1 , V nC1 ; W nC1 are solutions of the following linear system 8 ˆ ˆ
0; TO0 ; otherwise,

(3.55)

where the values of R0 and T0 are identical to the ones in the previous 2D simulations. The initial profile represents a small spherical crystal nucleus. We still use the fourfold anisotropic entropy coefficient ".r!/ in (2.4) and the default parameters in (3.52). Figure 12 shows some snapshots of phase variable ! at various time, that are consistent to the 2D simulations in Figure 2. To better visualize the interior structure of the dendritic cyrstal and temperature distribution, several 2D slices are presented in Figure 13. The 3D solutions show similar features to those obtained in [4–6, 10].

Figure 12. Time evolution of the dendritic crystal growth process in 3D using ıt D 0:01. The anisotropic coefficient ".r!/ is specified in (2.4), and all order parameters are from (3.52). Snapshots of the phase variable ! are taken at t D 0; 100; 200; 250; 300; 400; 500; 600; and 650.

Figure 13. The profile of the 2D slices for the temperature field T for the 3D dendritic crystal growth process of Figure 12 (a) and (b): snapshots of T at t D 200 and 500, respectively. (c): snapshots of the 2D slice of T .x; y; ´/j´D200 at t D 500. Copyright © 2016 John Wiley & Sons, Ltd.



5. CONCLUDING REMARKS In this paper, we have developed two efficient, time-stepping, first and second order scheme that are unconditionally energy stable for solving the PF–DCG model based on a novel IEQ approach. The proposed schemes bypass the difficulties encountered in the convex splitting and the stabilized approach. The developed schemes (i) are accurate (up to second order in time); (ii) are unconditionally energy stable; and (iii) are easy to implement (one only solves linear equations at each time step). Moreover, the semi-discrete linear system is positive definite so that one can apply any Krylov subspace methods with mass lumping as pre-conditioners to solve it efficiently. The new schemes are the first such linear and accurate schemes with unconditional energy stability for the PF–DCG model. The method is general enough to be extended to develop linear schemes for a large class of gradient flow problems with complex nonlinearities in the free-energy density. Although we consider only time discrete schemes in this study, the results can be carried over to any consistent finite-dimensional Galerkin approximations because the proofs are all based on a variational formulation with all test functions in the same space as the space of the trial functions. ACKNOWLEDGEMENTS

The authors would like to thank the anonymous reviewers for their suggestive and insightful reviews, which helped us to improve the quality of the paper.Jia Zhao is partially supported by an ASPIRE grant from the Office of the Vice President for Research at the University of South Carolina. Xiaofeng Yang is partially supported by the U.S. National Science Foundation under grant numbers DMS-1200487 and DMS-1418898. Qi Wang is partially supported by NSF-DMS-1200487 and DMS-1517347, NIH-2R01GM078994-05A1, SC EPSCOR/IDEA award, and NSFC-11571032. REFERENCES 1. Chen CC, Tsai YL, Lan CW. Adaptive phase field simulation of dendritic crystal growth in a forced flow: 2d vs. 3d morphologies. International Journal of HeatMass Transfer 2009; 52:1158–1166. 2. Halperin BI, Hohenberg PC, Ma SK. Renormalization-group methods for critical dynamics: i. recursion relations and effects of energy conservation. Physical Review 1974; B10:139–153. 3. Karma A, Rappel W. Quantitative phase-field modeling of dendritic growth in two and three dimensions. Physical Review E 1998; 57:4323–4349. 4. Karma A, Rappel W. Phase-field model of dendritic side branching with thermal noise. Physical Review E 1999; 60:3614–3625. 5. Li S, Lowengrub JS, Leo PH. Nonlinear morphological control of growing crystals. Physics D 2005; 208:209–219. 6. Li Y, Lee H, Kim J. A fast robust and accurate operator splitting method for phase field simulations of crystal growth. Journal of Crystal Growth 2011; 321:176–182. 7. Meca E, Shenoy V, Lowengrub J. Phase field modeling of two dimensional crystal growth with anisotropic diffusion. Physical Review E 2013; 88:052409. 8. Schulze TP. Simulation of dendritic growth into an undercooled melt using kinetic Monte Carlo techniques. Physical Review E 2008; 78:020601. 9. Plapp M, Karma A. Multiscale finite-difference-diffusion-Monte-Carlo method for simulating dendritic solidification. Journal of Computational Physics 2000; 165:592–619. 10. Shah A, Haider A, Shah S. Numerical simulation of two dimensional dendritic growth using phase field model. World Journal of Mechanics 2014; 4:128–136. 11. Tryggvason G, Bunner B, Esmaeeli A, Juric D, Al-Rawahi N, Tauber W, Han J, Nas S, Jan Y-J. A front-tracking method for the computations of multiphase flow. Journal of Computational Physics 2001; 169:708–759. 12. Warren JA, Boettinger WJ. Prediction of dentric growth and microsegregation patterns in a binary alloy using the phase field method. Acta Metallurgica et Materialia 1995; 43:689–703. 13. Jeong J-H, Goldenfeld N, Dantzig JA. Phase field model for three-dimensional dendritic growth with fluid flow. Physical Review E 2001; 55:041602. 14. Li Y, Kim J. Phase-field simulations of crystal growth with adaptive mesh refinement. International Journal of Heat and Mass Transfer 2012; 55:7926–7932. 15. Nestler B, Danilov D, Galenko P. Crystal growth of pure substances: phase field simulations in comparison with analytical and experimental results. Journal of Computational Physics 2005; 207:221–239. 16. Provatas N, Goldenfeld N, Dantzig J. Efficient computation of dendritic microstructures using adaptive mesh refinement. Physical Review Letter 1998; 80:3308–3311. 17. Ramirez JC, Beckermann C, Karma As, Diepers H-J. Phase-field modeling of binary alloy solidication with coupled heat and solute diffusion. Physical Review E 2004; 69:051607. Copyright © 2016 John Wiley & Sons, Ltd.


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