Controlling the accuracy of unconditionally stable algorithms in the Cahn-Hilliard equation

Cheng, Mowei; Warren, James A.

doi:10.1103/physreve.75.017702

Cited by 17 publications

(23 citation statements)

References 14 publications

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“…In addition, practical implementations in phase field modelling appears to produce qualitatively correct features, such as the development and propagation of interfaces. On the other hand, it has been widely observed that the convexity-splitting scheme can have large temporal errors [9,18,19]. The purpose of this paper is to improve the accuracy of the convexity splitting approach while retaining all of its other benefits.…”

Section: Introductionmentioning

confidence: 99%

Improving the accuracy of convexity splitting methods for gradient flow equations

Glasner

Orizaga

2016

Journal of Computational Physics

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This paper introduces numerical time discretization methods which significantly improve the accuracy of the convexity-splitting approach of Eyre (Unconditionally gradient stable time marching the Cahn-Hilliard equation, MRS Proceedings, vol. 529, 1998), while retaining the same numerical cost and stability properties. A first order method is constructed by iteration of a semi-implicit method based upon decomposing the energy into convex and concave parts. A second order method is also presented based on backwards differentiation formulas. Several extrapolation procedures for iteration initialization are proposed. We show that, under broad circumstances, these methods have an energy decreasing property, leading to good numerical stability. The new schemes are tested using two evolution equations commonly used in materials science: the Cahn-Hilliard equation and the phase field crystal equation. We find that our methods can increase accuracy by many orders of magnitude in comparison to the original convexitysplitting algorithm. In addition, the optimal methods require little or no iteration, making their computation cost similar to the original algorithm.

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Section: Introductionmentioning

confidence: 99%

Improving the accuracy of convexity splitting methods for gradient flow equations

Glasner

Orizaga

2016

Journal of Computational Physics

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“…In this case speed-ups of two orders of magnitude were obtained. Proofs of the numerical method are given in Cheng and Warren [257].…”

Section: The Phase Field Crystal Methods (Pfc)mentioning

confidence: 99%

The phase field technique for modeling multiphase materials

2008

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This paper reviews methods and applications of the phase field technique, one of the fastest growing areas in computational materials science. The phase field method is used as a theory and computational tool for predictions of the evolution of arbitrarily shaped morphologies and complex microstructures in materials. In this method, the interface between two phases (e.g. solid and liquid) is treated as a region of finite width having a gradual variation of different physical quantities, i.e. it is a diffuse interface model. An auxiliary variable, the phase field or order parameter φ( x), is introduced, which distinguishes one phase from the other. Interfaces are identified by the variation of the phase field. We begin with presenting the physical background of the phase field method and give a detailed thermodynamical derivation of the phase field equations. We demonstrate how equilibrium and non-equilibrium physical phenomena at the phase interface are incorporated into the phase field methods. Then we address in detail dendritic and directional solidification of pure and multicomponent alloys, effects of natural convection and forced flow, grain growth, nucleation, solid-solid phase transformation and highlight other applications of the phase field methods. In particular, we review the novel phase field crystal model, which combines atomistic length scales with diffusive time scales. We also discuss aspects of quantitative phase field modeling such as thin interface asymptotic analysis and coupling to thermodynamic databases. The phase field methods result in a set of partial differential equations, whose solutions require time-consuming large-scale computations and often limit the applicability of the method. Subsequently, we review numerical approaches to solve the phase field equations and present a finite difference discretization of the anisotropic Laplacian operator.

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“…For this benchmark problem, we study the effect that different implicit time integrators have on the simulation results. Explicit time integration methods enforce a maximum dt as a function of spatial discretization, dx, due to numerical instability; generally it is assumed that integration error is small because of this limitation [69]. However, many implicit time integrators are unconditionally stable, removing the naive link between dt and dx; larger time step sizes may be taken at the risk of unacceptably large integration error.…”

Section: Solidification and Dendritic Growthmentioning

confidence: 99%

Phase field benchmark problems for dendritic growth and linear elasticity

Jokisaari

Voorhees

Guyer

et al. 2018

Computational Materials Science

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We present the second set of benchmark problems for phase field models that are being jointly developed by the Center for Hierarchical Materials Design (CHiMaD) and the National Institute of Standards and Technology (NIST) along with input from other members in the phase field community. As the integrated computational materials engineering (ICME) approach to materials design has gained traction, there is an increasing need for quantitative phase field results. New algorithms and numerical implementations increase computational capabilities, necessitating standard problems to evaluate their impact on simulated microstructure evolution as well as their computational performance. We propose one benchmark problem for solidification and dendritic growth in a single-component system, and one problem for linear elasticity via the shape evolution of an elastically constrained precipitate. We demonstrate the utility and sensitivity of the benchmark problems by comparing the results of 1) dendritic growth simulations performed with different time integrators and 2) elastically constrained precipitate simulations with different precipitate sizes, initial conditions, and elastic moduli. These numerical benchmark problems will provide a consistent basis for evaluating different algorithms, both existing and those to be developed in the future, for accuracy and computational efficiency when applied to simulate physics often incorporated in phase field models.

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Controlling the accuracy of unconditionally stable algorithms in the Cahn-Hilliard equation

Cited by 17 publications

References 14 publications

Improving the accuracy of convexity splitting methods for gradient flow equations

Improving the accuracy of convexity splitting methods for gradient flow equations

The phase field technique for modeling multiphase materials

Phase field benchmark problems for dendritic growth and linear elasticity

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