Motion by mean curvature of curves on surfaces using the Allen-Cahn equation

Yongho Choi; Darae Jeong; Seunggyu Lee; Minhyun Yoo; Junseok Kim

doi:10.1016/j.ijengsci.2015.10.002

Motion by mean curvature of curves on surfaces using the Allen-Cahn equation

Yongho Choi, Darae Jeong, Seunggyu Lee, Minhyun Yoo, Junseok Kim

Department of Mathematics

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

27 Citations (Scopus)

Abstract

In this paper we develop a fast and accurate numerical method for motion by mean curvature of curves on a surface in three-dimensional space using the Allen-Cahn equation. We use a narrow band domain. We adopt a hybrid explicit numerical method which is based on an operator splitting method. First, we solve the heat equation by using an explicit standard Cartesian finite difference scheme. For the domain boundary cells, we use an interpolation using the closest point method. Then, we update the solution by using a closed-form solution. The proposing numerical algorithm is computationally efficient since we use a hybrid explicit numerical scheme and solve the governing equation only on the narrow domain. We perform a series of numerical experiments. The computational results are consistent with known analytic solutions.

Original language	English
Pages (from-to)	126-132
Number of pages	7
Journal	International Journal of Engineering Science
Volume	97
DOIs	https://doi.org/10.1016/j.ijengsci.2015.10.002
Publication status	Published - 2015 Dec 1

Bibliographical note

Funding Information:
The author (D. Jeong) was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology ( 2014R1A6A3A01009812 ). The corresponding author (J.S. Kim) was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIP) ( NRF-2014R1A2A2A01003683 ). The authors are grateful to the anonymous referees, whose valuable suggestions and comments significantly improved the quality of this paper.

Publisher Copyright:
© 2015 Elsevier Ltd. All rights reserved.

Keywords

Allen-Cahn equation
Closest point method
Hybrid explicit scheme
Motion by mean curvature
Narrow band domain

ASJC Scopus subject areas

General Materials Science
General Engineering
Mechanics of Materials
Mechanical Engineering

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10.1016/j.ijengsci.2015.10.002

Cite this

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title = "Motion by mean curvature of curves on surfaces using the Allen-Cahn equation",

abstract = "In this paper we develop a fast and accurate numerical method for motion by mean curvature of curves on a surface in three-dimensional space using the Allen-Cahn equation. We use a narrow band domain. We adopt a hybrid explicit numerical method which is based on an operator splitting method. First, we solve the heat equation by using an explicit standard Cartesian finite difference scheme. For the domain boundary cells, we use an interpolation using the closest point method. Then, we update the solution by using a closed-form solution. The proposing numerical algorithm is computationally efficient since we use a hybrid explicit numerical scheme and solve the governing equation only on the narrow domain. We perform a series of numerical experiments. The computational results are consistent with known analytic solutions.",

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author = "Yongho Choi and Darae Jeong and Seunggyu Lee and Minhyun Yoo and Junseok Kim",

note = "Funding Information: The author (D. Jeong) was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology ( 2014R1A6A3A01009812 ). The corresponding author (J.S. Kim) was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIP) ( NRF-2014R1A2A2A01003683 ). The authors are grateful to the anonymous referees, whose valuable suggestions and comments significantly improved the quality of this paper. Publisher Copyright: {\textcopyright} 2015 Elsevier Ltd. All rights reserved.",

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AU - Choi, Yongho

AU - Jeong, Darae

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AU - Yoo, Minhyun

AU - Kim, Junseok

N1 - Funding Information: The author (D. Jeong) was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology ( 2014R1A6A3A01009812 ). The corresponding author (J.S. Kim) was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIP) ( NRF-2014R1A2A2A01003683 ). The authors are grateful to the anonymous referees, whose valuable suggestions and comments significantly improved the quality of this paper. Publisher Copyright: © 2015 Elsevier Ltd. All rights reserved.

PY - 2015/12/1

Y1 - 2015/12/1

N2 - In this paper we develop a fast and accurate numerical method for motion by mean curvature of curves on a surface in three-dimensional space using the Allen-Cahn equation. We use a narrow band domain. We adopt a hybrid explicit numerical method which is based on an operator splitting method. First, we solve the heat equation by using an explicit standard Cartesian finite difference scheme. For the domain boundary cells, we use an interpolation using the closest point method. Then, we update the solution by using a closed-form solution. The proposing numerical algorithm is computationally efficient since we use a hybrid explicit numerical scheme and solve the governing equation only on the narrow domain. We perform a series of numerical experiments. The computational results are consistent with known analytic solutions.

AB - In this paper we develop a fast and accurate numerical method for motion by mean curvature of curves on a surface in three-dimensional space using the Allen-Cahn equation. We use a narrow band domain. We adopt a hybrid explicit numerical method which is based on an operator splitting method. First, we solve the heat equation by using an explicit standard Cartesian finite difference scheme. For the domain boundary cells, we use an interpolation using the closest point method. Then, we update the solution by using a closed-form solution. The proposing numerical algorithm is computationally efficient since we use a hybrid explicit numerical scheme and solve the governing equation only on the narrow domain. We perform a series of numerical experiments. The computational results are consistent with known analytic solutions.

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KW - Closest point method

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Motion by mean curvature of curves on surfaces using the Allen-Cahn equation

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

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