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

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

Research output: Contribution to journal › Article › peer-review

16 Citations (Scopus)

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.

Original language	English
Pages (from-to)	412-430
Number of pages	19
Journal	Communications in Computational Physics
Volume	27
Issue number	2
DOIs	https://doi.org/10.4208/cicp.OA-2018-0202
Publication status	Published - 2020

Bibliographical note

Funding Information:
The authors thank the reviewers for their constructive and helpful comments on the revision of this article. The first author (D. Jeong) was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIP) (NRF-2017R1E1A1A03070953). Y.B. Li is supported by National Natural Science Foundation of China (No.11601416, No.11631012). The corresponding author (J.S. Kim) was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2016R1D1A1B03933243).

Publisher Copyright:
© 2020 Global-Science Press.

Keywords

Cahn-Hilliard equation
Closest point method
Mass correction scheme
Narrow band domain

ASJC Scopus subject areas

Physics and Astronomy (miscellaneous)

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10.4208/cicp.OA-2018-0202

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title = "A conservative numerical method for the Cahn-Hilliard equation with generalized mobilities on curved surfaces in three-dimensional space",

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, Closest point method, Mass correction scheme, Narrow band domain",

author = "Darae Jeong and Yibao Li and Chaeyoung Lee and Junxiang Yang and Junseok Kim",

note = "Funding Information: The authors thank the reviewers for their constructive and helpful comments on the revision of this article. The first author (D. Jeong) was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIP) (NRF-2017R1E1A1A03070953). Y.B. Li is supported by National Natural Science Foundation of China (No.11601416, No.11631012). The corresponding author (J.S. Kim) was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2016R1D1A1B03933243). Publisher Copyright: {\textcopyright} 2020 Global-Science Press.",

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AU - Kim, Junseok

N1 - Funding Information: The authors thank the reviewers for their constructive and helpful comments on the revision of this article. The first author (D. Jeong) was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIP) (NRF-2017R1E1A1A03070953). Y.B. Li is supported by National Natural Science Foundation of China (No.11601416, No.11631012). The corresponding author (J.S. Kim) was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2016R1D1A1B03933243). Publisher Copyright: © 2020 Global-Science Press.

PY - 2020

Y1 - 2020

N2 - 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.

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.

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A conservative numerical method for the Cahn-Hilliard equation with generalized mobilities on curved surfaces in three-dimensional space

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