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Volume 27, Issue 2
A Conservative Numerical Method for the Cahn–Hilliard Equation with Generalized Mobilities on Curved Surfaces in Three-Dimensional Space

Darae Jeong, Yibao Li, Chaeyoung Lee, Junxiang Yang & Junseok Kim

DOI: 10.4208/cicp.OA-2018-0202

Commun. Comput. Phys., 27 (2020), pp. 412-430.

Published online: 2019-12

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  • Abstract

In this paper, we develop a conservative numerical method for the Cahn– Hilliard equation with generalized mobilities on curved surfaces in three-dimensional space. We use an unconditionally gradient stable nonlinear splitting numerical scheme and solve the resulting system of implicit discrete equations on a discrete narrow band domain by using a Jacobi-type iteration. For the domain boundary cells, we use the trilinear interpolation using the closest point method. The proposing numerical algorithm is computationally efficient because we can use the standard finite difference Laplacian scheme on three-dimensional Cartesian narrow band mesh instead of discrete Laplace–Beltrami operator on triangulated curved surfaces. In particular, we employ a mass conserving correction scheme, which enforces conservation of total mass. We perform numerical experiments on the various curved surfaces such as sphere, torus, bunny, cube, and cylinder to demonstrate the performance and effectiveness of the proposed method. We also present the dynamics of the CH equation with constant and space-dependent mobilities on the curved surfaces.

  • Keywords

Cahn–Hilliard equation, mass correction scheme, narrow band domain, closest point method.

  • AMS Subject Headings

37M05, 65M06, 65M22, 65P99

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

tinayoyo@kangwon.ac.kr (Darae Jeong)

yibaoli@xjtu.edu.cn (Yibao Li)

chae1228@korea.ac.kr (Chaeyoung Lee)

nexusxiang@outlook.com (Junxiang Yang)

cfdkim@korea.ac.kr (Junseok Kim)

  • BibTex
  • RIS
  • TXT
@Article{CiCP-27-412, author = {Jeong , DaraeLi , YibaoLee , ChaeyoungYang , Junxiang and Kim , Junseok}, title = {A Conservative Numerical Method for the Cahn–Hilliard Equation with Generalized Mobilities on Curved Surfaces in Three-Dimensional Space}, journal = {Communications in Computational Physics}, year = {2019}, volume = {27}, number = {2}, pages = {412--430}, abstract = {

In this paper, we develop a conservative numerical method for the Cahn– Hilliard equation with generalized mobilities on curved surfaces in three-dimensional space. We use an unconditionally gradient stable nonlinear splitting numerical scheme and solve the resulting system of implicit discrete equations on a discrete narrow band domain by using a Jacobi-type iteration. For the domain boundary cells, we use the trilinear interpolation using the closest point method. The proposing numerical algorithm is computationally efficient because we can use the standard finite difference Laplacian scheme on three-dimensional Cartesian narrow band mesh instead of discrete Laplace–Beltrami operator on triangulated curved surfaces. In particular, we employ a mass conserving correction scheme, which enforces conservation of total mass. We perform numerical experiments on the various curved surfaces such as sphere, torus, bunny, cube, and cylinder to demonstrate the performance and effectiveness of the proposed method. We also present the dynamics of the CH equation with constant and space-dependent mobilities on the curved surfaces.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.OA-2018-0202}, url = {http://global-sci.org/intro/article_detail/cicp/13452.html} }
TY - JOUR T1 - A Conservative Numerical Method for the Cahn–Hilliard Equation with Generalized Mobilities on Curved Surfaces in Three-Dimensional Space AU - Jeong , Darae AU - Li , Yibao AU - Lee , Chaeyoung AU - Yang , Junxiang AU - Kim , Junseok JO - Communications in Computational Physics VL - 2 SP - 412 EP - 430 PY - 2019 DA - 2019/12 SN - 27 DO - http://doi.org/10.4208/cicp.OA-2018-0202 UR - https://global-sci.org/intro/article_detail/cicp/13452.html KW - Cahn–Hilliard equation, mass correction scheme, narrow band domain, closest point method. AB -

In this paper, we develop a conservative numerical method for the Cahn– Hilliard equation with generalized mobilities on curved surfaces in three-dimensional space. We use an unconditionally gradient stable nonlinear splitting numerical scheme and solve the resulting system of implicit discrete equations on a discrete narrow band domain by using a Jacobi-type iteration. For the domain boundary cells, we use the trilinear interpolation using the closest point method. The proposing numerical algorithm is computationally efficient because we can use the standard finite difference Laplacian scheme on three-dimensional Cartesian narrow band mesh instead of discrete Laplace–Beltrami operator on triangulated curved surfaces. In particular, we employ a mass conserving correction scheme, which enforces conservation of total mass. We perform numerical experiments on the various curved surfaces such as sphere, torus, bunny, cube, and cylinder to demonstrate the performance and effectiveness of the proposed method. We also present the dynamics of the CH equation with constant and space-dependent mobilities on the curved surfaces.

Jeong , DaraeLi , YibaoLee , ChaeyoungYang , Junxiang and Kim , Junseok. (2019). A Conservative Numerical Method for the Cahn–Hilliard Equation with Generalized Mobilities on Curved Surfaces in Three-Dimensional Space. Communications in Computational Physics. 27 (2). 412-430. doi:10.4208/cicp.OA-2018-0202
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