A Time Integration Method for PhaseField Modeling
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Abstract
A novel numerical time integration for solving phasefield problems is presented. This method includes the generalized single step single solve (GSSSS) family of algorithms, which can preserve secondorder time accuracy as well as provide controllable numerical dissipation independently on each time variable. Furthermore, we demonstrate an enhancement of the time integration method that can reduce numerical oscillation of differential algebraic equation from phasefield model. The algebraic equation is evaluated at \(t_{n+1}\) instead of the general time level \(t_{n+W_{1}}\). The enhancement reduces the numerical oscillation due to the nondissipative scheme. Two popular phasefield examples, the Cahn–Hilliard equations and simple phasefieldcrystal equations, are used to demonstrate the capability of proposed time integration scheme. We conclude that this approach has a significant advantage over currently used algorithms, and provides a new avenue for robustness of time integration schemes for phasefield problems with a highorder term in free energy.
Keywords
Cahn–Hilliard model Differential algebraic equations GSSSS family of algorithms Phasefieldcrystal model Phasefield methodIntroduction
Numerical simulation of phasefield modelling has become popular in recent years. It was originally applied to solidification dynamics [3], and is now widely implemented in various largescale problems with interfacial phenomena such as fracture dynamics [15], bubble kinematics [12], and microstructure evolution [5]. The phasefield method uses a partial differential equation, known as the order parameter, to describe the phase transition at an interface. Two specific values (e.g. − 1 and 1) represent each side of the phase with a continuous diffusion boundary. Thus, the fronttracking technique is no longer necessary and the phase propagation can then be more easily traced [6, 17]. The phasefield method is advancing in solving problems which have difficulties in experiments, and thus numerical simulation of the phasefield method is continuing to improve in terms of robustness and accuracy in the pursuit of a better understanding of complex interfacial phenomena [6, 7, 8, 9, 12, 18, 22].
An essential feature arising in the phasefield method is that phasefield equations can be derived from a higherorder energetic variational formulation [10]. This variational form usually contains a biharmonic (or higher) operator. Therefore, numerical method such as finite element methods require higher continuity (\(C^{1}\) or higher) to approximate the solution of the weak form. Another way to bypass using higher continuity function is to introduce extra variables to represent these highorder terms. Thus linear piecewise approximation is applicable for the weak form solution. This procedure for the phasefield method is very common [23] and turns phasefield equations from a differential system to a differential algebraic system with index1. Another noteworthy feature is that the phasefield method often results in a stiff system according to the limit of the interface width. As the width reduces, the system becomes extremely stiff [23], and usually cause numerical instability. Also, in the formulation of the phasefield model, thermal noise are usually introduced at interface to obtain certain feature, such as dendrite morphology[23], which is significant in physics.
Semiimplicit time integration schemes were first used for phasefield equations, meaning they solve differential equations and algebraic equations separately with different method[2, 16, 24], for example, Euler method with Crank–Nicolson method. This approach may reduce global time accuracy, and is conditionally stable. Also, solving DAEs separately is computational inefficient. To address this issue, implicit schemes with numerical dissipation are provided as a solution. Computer softwares with dissipative scheme such as, DASSL [21] or ODE15s in MATLAB, has been working well in simulating DAEs. The backward difference formulation (BDF) and some RungeKutta type methods are also widely used for phasefield problems [1, 29]. However, these methods demonstrate remarkable numerical dissipation, which may oversmear the thermal fluctuation mentioned above. Investigation of nondissipative scheme and controllable dissipation with unconditionally stability in phasefield modeling is required.
As a result, we introduce a secondorder accurate, unconditionally stable, single step single solve scheme. The proposed algorithm is capable of control numerical dissipation separately on the prime variables and their time derivative. In addition, we evaluate the algebraic variables at timelevel \(t_{n+1}\), which can stabilize the numerical solution when using midpointtype nondissipative methods included in GSSSS. This extended approach demonstrates a possible avenue for discovering robust time integration schemes. In “The framework of GSSSS family of algorithms for phasefield equations”, the framework of the GSSSS family of algorithms is defined for index1 DAEs. In “Stability and accuracy of GSSSS family of algorithm”, we briefly introduce the features of GSSSS in stability and accuracy. In “Numerical examples”, two numerical examples are shown to verify analytical algorithmic properties and the extension to nonlinear problems. Finally, in “Conclusions”, we give the conclusion of this study.
The Framework of GSSSS Family of Algorithms for PhaseField Equations
Notations used for solving DAEs with GSSSS time integration method
Notation  Definitions 

\(\varOmega\), \(\varGamma\)  Domain and boundary of the phase field problem 
\(\partial\)  Partial differential operator 
\(\delta\)  Variation differential operator 
\(\nabla\)  Gradient operator 
\(\nabla ^2\)  Laplacian operator \(\nabla ^2= \nabla \cdot \nabla\) 
\(\theta ^{h}\)  Galerkin finite element approximation of \(\theta\) 
\(\varDelta t\)  Timestep size in fully discretized system of DAEs 
\(\dot{\theta }\)  Time derivative of variable \(\theta\) 
\(\theta _{0}\), \(t_{0}\)  Variable \(\theta\) and time t evaluated at zero timestep \(t=0\) 
\(\theta_{n}\), \(t_{n}\)  Variable \(\theta\) and time t evaluated at n timestep \(t=t_{n}\) 
\(t_{n+W_{1}}\)  Time t evaluated at \(n+W_{1}\) timestep: \(t=(1W_{1})t_{n}+W_{1}t_{n+1}\) 
\(\tilde{\theta }\)  Variable \(\theta\) evaluated at time \(t=t_{n+W_{1}}\) where for displacement, velocity and algebraic variables are defined in Eq. (7) 

Unconditional stability

Secondorder time accuracy

Controllable numerical dissipation by \(\rho\) and \(\rho _s\)

Zeroorder overshoot behaviour

Principle root \(\rho\) controls the numerical dissipation on \(\phi ^{h}\)

Spurious root \(\rho _s\) controls the numerical dissipation on \(\dot{\phi ^{h}}\)
Options of evaluation timelevel for DAEs
Option  Equations of DAE system  Timelevel 

Option I  \(\tilde{\dot{\phi ^{h}}} = F(\tilde{\phi ^{h}},\tilde{\psi ^{h}},t_{n+W_{1}})\)  \(t_{n+W_{1}}\) 
\(0 = Q(\tilde{\phi ^{h}},\tilde{\psi ^{h}},t_{n+W_{1}})\)  \(t_{n+W_{1}}\)  
Option II  \(\tilde{\dot{\phi ^{h}}} = F(\tilde{\phi ^{h}},\tilde{\psi ^{h}},t_{n+W_{1}})\)  \(t_{n+W_{1}}\) 
\(0 = Q(\phi ^{h}_{n+1},\psi ^{h}_{n+1},t_{n+1})\)  \(t_{n+1}\) 
Stability and Accuracy of GSSSS Family of Algorithm
In the general situation of different normalized wave number \(\theta = 0\) to \(\pi\), one will find out the spectral roots from Option I always contain one roots \(\xi ^{Option I}_{k}=\rho\), such that makes it hard to reduces numerical perturbation when \(\rho = 1\). On the other hand, this roots is removed in the case of Option II such that Option II always contain one roots \(\xi ^{Option II}_{k}=0\) with two conjugate roots, which making the solution more stable since it always satisfy the stability criteria (21). In this way, adapting Option II should be more robust in modeling phasefield equations.
Stability of GSSSS scheme for two options in index1 DAEs
Option  \((\rho ,\rho _{s})\)  Dissipation  Solution 

Option I  (1, 1)  No  Unstable 
\((\rho< 1,\rho _{s} < 1)\)  Yes  Stable  
Option II  (1, 1)  No  Stable 
\((\rho< 1,\rho _{s} < 1)\)  Yes  Stable 
It is shown in [25, 28] that the algorithmic parameter \(\rho\) controls the numerical dissipation on the prime variables \(\phi ^{h}\) and algebraic variables \(\psi ^{h}\). On the other hand, \(\rho _{s}\) controls the numerical dissipation on \(\dot{\phi ^{h}}\). Larger \(\rho\) and \(\rho _s\) provide less dissipation and \((\rho ,\rho _s)=(1,1)\) (equivalent to Crank–Nicolson scheme) produces a completely nondissipative system. Also, it should be noted that \(\rho =1\) produces the time integration scheme with timelevel \(t_{n+W_{1}}=t_{n+0.5}\), which is equivalent to the general midpoint schemes [25, 26]. From Table 3 and the temporal analysis, we can see that for nondissipative scheme, evaluating the algebraic equations at the end of the timelevel reproduce a stable solution by avoiding multiple spectral roots lying with unit length. On the other hand, adding numerical dissipation always gives a stable solution. In addition, it is proved in [25] under Option II, perturbation in algebraic variables will decay if \((\rho ,\rho _s)=(1,\rho _s)\) and \(\rho _s<1\). Related results will be shown in “Numerical examples”.
As noted in “The framework of GSSSS family of algorithms for phasefield equations”, all schemes in GSSSS family of algorithms are second order time accuracy. A noteworthy property of the GSSSS family of algorithms is the timelevel of the time derivative \(\dot{\phi ^{h}}\), which \(\dot{\phi ^{h}}\) preserve secondorder time accuracy when \(\dot{\phi ^{h}}\) is evaluated at time \(t_{n(\varLambda _{6}W_{1}W_{1})}\) instead of time \(t_{n}\). The complete proof of time order accuracy can be found in [19, 25, 26]. To conclude, the features of unconditional stability and second order accuracy is practical for solving differentialalgebraic phasefield equations with perturbed initial conditions or noises. Numerical evidence will be provided to verify such properties in the next section.
Numerical Examples
Two numerical examples are used to demonstrated and explored the option of the proposed scheme. The first example given is the Cahn–Hilliard equation described in [23], which expresses the separation of mixtures with a miscibility interface in the phase diagram, and is probably the most well known phasefield model. The second example is a simple phasefieldcrystal model (PFC) as in [29] which is an extension of the phasefield model. PFC has recently become popular in multiscale modelling of microstructure evolution. Both examples are classic problems in phasefield simulation. In addition, they can be derived from highorder free energy, which is common in this field. To discretise in space, we employ the Galerkin finite element method with \(C^{0}\) elements in a regular mesh. Then, we apply the GSSSS family of algorithms to demonstrate the time marching results.
The solution is produced under several different combinations of \(\rho\) and \(\rho _{s}\) resulting in an algorithms denoted by \((\rho ,\rho _{s})\). For both cases, convergence is plotted using a numerical solution with very refined timesteps, which is treated as the exact solution \(z_{exact}\). Then, the comparative numerical solution is produced by larger timesteps, \(z_{numerical}\) and the definition of error is given by \((z_{numerical}z_{exact})/z_{exact}\).
Cahn–Hilliard Equations
Finally, a convergence plot for the Cahn–Hilliard equation is given in Fig. 6. The secondorder accuracy of the GSSSS scheme can be seen for different choices of \((\rho ,\rho _s)\).
PhaseFieldCrystal Equations
To see how perturbation influences the solution, a perturbed initial condition (randomly distributed for \(\bar{\phi }\in [0.05,0.05]\)) is also used for the phasefieldcrystal model as shown in Fig. 7a–f. With such initial condition, the numerical instability can be shown more clearly in Fig. 9c, d.
Finally, a convergence plot for the PFC model can be found in Fig. 11. The secondorder accuracy of the GSSSS scheme can be seen for different choices of \((\rho ,\rho _s)\). For both numerical examples, the PF and PFC model verify the numerical properties for stability and accuracy under the different Options. Option I acts as an initial and straightforward approach, and shows unstable solutions without introducing numerical dissipation. Even if we artificially add the dissipation to reduce the oscillation, it takes a while for the solution to converge. On the other hand, the new approach, Option II, shows stable and robust solutions and can deal with the instability arising in nondissipative midpoint methods. The proposed method had been applied on phasefiled modeling for manufacturing process shown in [13].
Conclusions
In this research, we presented an unconditionally stable, generalized single step single solve scheme (GSSSS) to solve phasefield problems in differential algebraic system. GSSSS can control numerical dissipation as well as preserve secondorder time accuracy for this problem. We also demonstrated an enhancement of the original approach (Option I) by evaluating the algebraic equation at timelevel \(t_{n+1}\) (Option II). This reduced the instability arising in the general phasefield problem, and saved unstable schemes, such as Crank–Nicolson and other midpoint methods, from oscillatory behaviour in algebraic equations. Thus, nondissipative schemes such as Crank–Nicolson or the Midpoint rule can be retained. Two numerical examples were provided as numerical evidence to verify: (1) the algorithmic accuracy and stability properties; (2) numerical dissipation is important for controlling numerical oscillation; and (3) Option II (evaluate the algebraic equation at timelevel \(t_{n+1}\)) is a valid way in time integration which can overcome instability arising in stiff DAEs, especially for phasefield problems with highorder free energy. This research demonstrates a new avenue for nondissipative schemes and provides an opportunities toward robust time integration scheme for phasefield modelling.
Notes
Acknowledgements
We would like to acknowledge the help regarding GSSSS time integration method from Dr. Shimada and Prof. Tamma at University of Minnesota, Twin Cities. We are grateful for the computational resources from National Center of Highperformance Computing (NCHC). This work is supported by the Industrial Technology Research Institute (ITRI) at Hsinchu, Taiwan.
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