An adaptive time-stepping strategy for solving the phase field crystal model

Zhang, Zhengru; Ma, Yuan; Qiao, Zhonghua

doi:10.1016/j.jcp.2013.04.031

Cited by 109 publications

(72 citation statements)

References 23 publications

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“…Therefore, efficient numerical solutions for these problems usually involve adaptive time stepping schemes. Different strategies have already been proposed, such as the ones found in [21,27] for the Allen-Cahn and Cahn-Hilliard equations and in [63] for the phase-field crystal equation. Even though the methods presented therein successfully decrease the computational time taken to reach steady state solutions, they still have room for considerable computational savings.…”

Section: Time Adaptivitymentioning

confidence: 99%

“…Even though shown to be robust and used since in other works [23,35,39], this strategy is inefficient given that the solution must be computed twice at each time step. In [27,63], no recovery strategies are proposed when the numerical solver fails. This implies a period of trial and error is necessary to tune the solver parameters for the specific equation being solved.…”

Section: Time Adaptivitymentioning

confidence: 99%

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An energy-stable time-integrator for phase-field models

Vignal

Collier

Dalcín

et al. 2017

Computer Methods in Applied Mechanics and Engineering

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We introduce a provably energy-stable time-integration method for general classes of phase-field models with polynomial potentials. We demonstrate how Taylor series expansions of the nonlinear terms present in the partial differential equations of these models can lead to expressions that guarantee energy-stability implicitly, which are second-order accurate in time. The spatial discretization relies on a mixed finite element formulation and isogeometric analysis. We also propose an adaptive time-stepping discretization that relies on a first-order backward approximation to give an error-estimator. This error estimator is accurate, robust, and does not require the computation of extra solutions to estimate the error. This methodology can be applied to any second-order accurate time-integration scheme. We present numerical examples in two and three spatial dimensions, which confirm the stability and robustness of the method. The implementation of the numerical schemes is done in PetIGA, a high-performance isogeometric analysis framework.

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Section: Time Adaptivitymentioning

confidence: 99%

Section: Time Adaptivitymentioning

confidence: 99%

An energy-stable time-integrator for phase-field models

Vignal

Collier

Dalcín

et al. 2017

Computer Methods in Applied Mechanics and Engineering

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“…This energy stability is not enough to guarantee the uniform boundedness of φ k ∞ since the appearance of the term 1 4 ((φ n ) 2 , (φ n+1 ) 2 ). In this sense, the energy stability of Theorem 3.1 is weaker than that for the two level nonlinear difference scheme in [15].…”

Section: Remark 32mentioning

confidence: 93%

“…Wang and Wise [13] constructed a difference scheme with the convergence order one in time for the modified phase field crystal equation. Zhang et al [15] established the following difference scheme of second order in time:…”

Section: Introductionmentioning

confidence: 99%

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Two finite difference schemes for the phase field crystal equation

Cao

Sun

2015

Sci. China Math.

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The phase field crystal (PFC) model is a nonlinear evolutionary equation that is of sixth order in space. In the first part of this work, we derive a three level linearized difference scheme, which is then proved to be energy stable, unique solvable and second order convergent in L 2 norm by the energy method combining with the inductive method. In the second part of the work, we analyze the unique solvability and convergence of a two level nonlinear difference scheme, which was developed by Zhang et al. in 2013. Some numerical results with comparisons are provided. Keywordsphase field crystal model, nonlinear evolutionary equation, finite difference scheme, solvability, convergence MSC(2010) 65M06, 65M12, 65M15Citation: Cao H Y, Sun Z Z. Two finite difference schemes for the phase field crystal equation.

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Multiphase Polarization in Ion‐Intercalation Nanofilms: General Theory Including Various Surface Effects and Memory Applications

Tian

Bazant

2023

Adv Funct Materials

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Ion concentration polarization (CP, current‐induced concentration gradient adjacent to a charge‐selective interface) has been well studied for single‐phase mixed conductors (e.g., liquid electrolyte), but multiphase CP has been rarely addressed in literature. In our recent publication, we proposed that CP above certain threshold currents can flip the phase distribution in multiphase ion‐intercalation nanofilms sandwiched by ion‐blocking electrodes. This phenomenon is known as multiphase polarization (MP). It is then proposed that MP can further lead to nonvolatile interfacial resistive switching (RS) for asymmetric electrodes with ion‐modulated electron transfer, which theory can reproduce the experimental results of LTO memristors. In this study, a comprehensive 2D phase‐field model is derived for coupled ion‐electron transport in ion‐intercalation materials, with surface effects including electron transfer kinetics, non‐neutral wetting, energy relaxation, and surface charge. Then, the model is used to study MP. Time evolution of phase boundaries is presented, and analyze the switching time, current, energy, and cyclic voltammetry, for various boundary conditions. It is found that the switching performance can be improved significantly by manipulating surface conditions and mean concentration. Finally, the prospects of MP‐based memories and possible extensions of the current model is discussed.

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An adaptive time-stepping strategy for solving the phase field crystal model

Cited by 109 publications

References 23 publications

An energy-stable time-integrator for phase-field models

An energy-stable time-integrator for phase-field models

Two finite difference schemes for the phase field crystal equation

Multiphase Polarization in Ion‐Intercalation Nanofilms: General Theory Including Various Surface Effects and Memory Applications

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