A simple and efficient finite difference method for the phase-field crystal equation on curved surfaces

Hyun Geun Lee, Junseok Kim

Research output: Contribution to journalArticle

12 Citations (Scopus)

Abstract

We present a simple and efficient finite difference method for the phase-field crystal (PFC) equation on curved surfaces embedded in R3. We employ a narrow band neighborhood of a curved surface that is defined as a zero level set of a signed distance function. The PFC equation on the surface is extended to the three-dimensional narrow band domain. By using the closest point method and applying a pseudo-Neumann boundary condition, we can use the standard seven-point discrete Laplacian operator instead of the discrete Laplace-Beltrami operator on the surface. The PFC equation on the narrow band domain is discretized using an unconditionally stable scheme and the resulting implicit discrete system of equations is solved by using the Jacobi iterative method. Computational results are presented to demonstrate the efficiency and usefulness of the proposed method.

Original languageEnglish
Pages (from-to)32-43
Number of pages12
JournalComputer Methods in Applied Mechanics and Engineering
Volume307
DOIs
Publication statusPublished - 2016 Aug 1

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curved surfaces
Finite difference method
crystal field theory
narrowband
Crystals
operators
Iterative methods
Boundary conditions
boundary conditions

Keywords

  • Closest point method
  • Curved surface
  • Finite difference method
  • Narrow band domain
  • Phase-field crystal equation

ASJC Scopus subject areas

  • Computer Science Applications
  • Computational Mechanics
  • Mechanics of Materials
  • Mechanical Engineering
  • Physics and Astronomy(all)

Cite this

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abstract = "We present a simple and efficient finite difference method for the phase-field crystal (PFC) equation on curved surfaces embedded in R3. We employ a narrow band neighborhood of a curved surface that is defined as a zero level set of a signed distance function. The PFC equation on the surface is extended to the three-dimensional narrow band domain. By using the closest point method and applying a pseudo-Neumann boundary condition, we can use the standard seven-point discrete Laplacian operator instead of the discrete Laplace-Beltrami operator on the surface. The PFC equation on the narrow band domain is discretized using an unconditionally stable scheme and the resulting implicit discrete system of equations is solved by using the Jacobi iterative method. Computational results are presented to demonstrate the efficiency and usefulness of the proposed method.",
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