An unconditionally energy-stable method for the phase field crystal equation

Gómez, Héctor; Nogueira, Xesús

doi:10.1016/j.cma.2012.03.002

Cited by 146 publications

(124 citation statements)

References 48 publications

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“…It is well known that Newton's method is the most popular method to solve the related nonlinear equations [15,34]. However, the standard Newton's method solves a system of nonlinear equations in a simultaneous and coupled way, but this way may be not perfect for the considered nonlinear system because it consists of one nonlinear partial differential equation and N − 1 nonlinear algebraic equations.…”

Section: A Modified Newton's Methods For Euler-lagrange Equationsmentioning

confidence: 99%

Numerical Methods for a Multicomponent Two-Phase Interface Model with Geometric Mean Influence Parameters

Kou¹,

Sun²

2015

SIAM J. Sci. Comput.

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Abstract. In this paper, we consider an interface model for multicomponent two-phase fluids with geometric mean influence parameters, which is popularly used to model and predict surface tension in practical applications. For this model, there are two major challenges in theoretical analysis and numerical simulation: the first one is that the influence parameter matrix is not positive definite; the second one is the complicated structure of the energy function, which requires us to find out a physically consistent treatment. To overcome these two challenging problems, we reduce the formulation of the energy function by employing a linear transformation and a weighted molar density, and furthermore, we propose a local minimum grand potential energy condition to establish the relation between the weighted molar density and mixture compositions. From this, we prove the existence of the solution under proper conditions and prove the maximum principle of the weighted molar density. For numerical simulation, we propose a modified Newton's method for solving this nonlinear model and analyze its properties; we also analyze a finite element method with a physicalbased adaptive mesh-refinement technique. Numerical examples are tested to verify the theoretical results and the efficiency of the proposed methods.

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Section: A Modified Newton's Methods For Euler-lagrange Equationsmentioning

confidence: 99%

Numerical Methods for a Multicomponent Two-Phase Interface Model with Geometric Mean Influence Parameters

Kou¹,

Sun²

2015

SIAM J. Sci. Comput.

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“…The force vector f is obtained by substituting the above manufactured solution into equations (32)- (33). The dimensionless parameters are fixed to be Re = 2.0×10 1 and We = 1.0×10 2 .…”

Section: Manufactured Solution For Code Verificationmentioning

confidence: 99%

“…It may be noted that the stability proof is valid for all η ≥ 0. If η = 0, the last term in (121) does not produce numerical dissipation [33]. Positive η provides controllable numerical dissipation through the value of C selected.…”

mentioning

confidence: 99%

Functional Entropy Variables: A New Methodology for Deriving Thermodynamically Consistent Algorithms for Complex Fluids, with Particular Reference to the Isothermal Navier-Stokes-Korteweg Equations

Liu¹,

Gómez²,

Evans³

et al. 2012

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Public reporting burden for the collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources, gathering and maintaining the data needed, and completing and reviewing the collection of information. Send comments regarding this burden estimate or any other aspect of this collection of information, including suggestions for reducing this burden, to Washington Headquarters Services, Directorate for Information Operations and Reports, 1215 Jefferson Davis Highway, Suite 1204, Arlington VA 22202-4302. Respondents should be aware that notwithstanding any other provision of law, no person shall be subject to a penalty for failing to comply with a collection of information if it does not display a currently valid OMB control number. We propose a new methodology for the numerical solution of the isothermal Navier-Stokes-Korteweg equations. Our methodology is based on a semi-discrete Galerkin method invoking functional entropy variables, a generalization of classical entropy variables, and a new time integration scheme. We show that the resulting fully discrete scheme is unconditionally stable-in-energy, second-order time-accurate, and mass-conservative. We utilize isogeometric analysis for spatial discretization and verify the aforementioned properties by adopting the method of manufactured solutions and comparing coarse mesh solutions with overkill solutions. Various problems are simulated to show the capability of the method. Our methodology provides a means of constructing unconditionally stable numerical schemes for nonlinear non-convex hyperbolic systems of conservation laws. AbstractWe propose a new methodology for the numerical solution of the isothermal Navier-StokesKorteweg equations. Our methodology is based on a semi-discrete Galerkin method invoking functional entropy variables, a generalization of classical entropy variables, and a new time integration scheme. We show that the resulting fully discrete scheme is unconditionally stable-in-energy, second-order time-accurate, and mass-conservative. We utilize isogeometric analysis for spatial discretization and verify the aforementioned properties by adopting the method of manufactured solutions and comparing coarse mesh solutions with overkill solutions. Various problems are simulated to show the capability of the method. Our methodology provides a means of constructing unconditionally stable numerical schemes for nonlinear non-convex hyperbolic systems of conservation laws.

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“…It is therefore of great importance to study scalable parallel algorithms for the PFC equation. Although numerical methods for the PFC equation have been investigated in a number of publications, e.g., [5], [14], [17], [26], [28], works dedicated to parallel algorithms are not yet to be seen. There are some successful studies on scalable parallel algorithms for some other phase-field problems such as the Cahn-Hilliard equation [29], [31] and the coupled Allen-Cahn/Cahn-Hilliard equations [24], [27], [30].…”

Section: Introductionmentioning

confidence: 99%

A Scalable Implicit Solver for Phase Field Crystal Simulations

Yang

Cai

2013

2013 IEEE International Symposium on Parallel &Amp; Distributed Processing, Workshops and PHD Forum

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Abstract-The phase field crystal equation has become a popular model for simulating micro-structures in materials science but is very computationally expensive to solve. A highly scalable solver for phase field crystal modeling is presented in this paper. The equation is discretized with a stabilized implicit finite difference method and the time step size is adaptively controlled to obtain physically meaningful solutions. The nonlinear system arising at each time step is solved by using a parallel Newton-Krylov-Schwarz algorithm. In order to achieve good performance, low-order homogeneous boundary conditions are imposed on subdomain interfaces in the Schwarz preconditioner. Experiments are carried out to exploit optimal choices of the preconditioner type, the subdomain solver and the overlap size. Numerical results are provided to show that the solver is scalable to thousands of processor cores.

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An unconditionally energy-stable method for the phase field crystal equation

Cited by 146 publications

References 48 publications

Numerical Methods for a Multicomponent Two-Phase Interface Model with Geometric Mean Influence Parameters

Numerical Methods for a Multicomponent Two-Phase Interface Model with Geometric Mean Influence Parameters

Functional Entropy Variables: A New Methodology for Deriving Thermodynamically Consistent Algorithms for Complex Fluids, with Particular Reference to the Isothermal Navier-Stokes-Korteweg Equations

A Scalable Implicit Solver for Phase Field Crystal Simulations

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