• Title/Summary/Keyword: Allen-Cahn equation

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A NUMERICAL METHOD FOR THE MODIFIED VECTOR-VALUED ALLEN-CAHN PHASE-FIELD MODEL AND ITS APPLICATION TO MULTIPHASE IMAGE SEGMENTATION

  • Lee, Hyun Geun;Lee, June-Yub
    • Journal of the Korean Society for Industrial and Applied Mathematics
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    • v.18 no.1
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    • pp.27-41
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    • 2014
  • In this paper, we present an efficient numerical method for multiphase image segmentation using a multiphase-field model. The method combines the vector-valued Allen-Cahn phase-field equation with initial data fitting terms containing prescribed interface width and fidelity constants. An efficient numerical solution is achieved using the recently developed hybrid operator splitting method for the vector-valued Allen-Cahn phase-field equation. We split the modified vector-valued Allen-Cahn equation into a nonlinear equation and a linear diffusion equation with a source term. The linear diffusion equation is discretized using an implicit scheme and the resulting implicit discrete system of equations is solved by a multigrid method. The nonlinear equation is solved semi-analytically using a closed-form solution. And by treating the source term of the linear diffusion equation explicitly, we solve the modified vector-valued Allen-Cahn equation in a decoupled way. By decoupling the governing equation, we can speed up the segmentation process with multiple phases. We perform some characteristic numerical experiments for multiphase image segmentation.

A NUMERICAL METHOD FOR SOLVING ALLEN-CAHN EQUATION

  • Huang, Pengzhan;Abduwali, Abdurishit
    • Journal of applied mathematics & informatics
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    • v.29 no.5_6
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    • pp.1477-1487
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    • 2011
  • We propose a numerical method for solving Allen-Cahn equation, in both one-dimensional and two-dimensional cases. The new scheme that is explicit, stable, and easy to compute is obtained and the proposed method provides a straightforward and effective way for nonlinear evolution equations.

HIGHER ORDER OPERATOR SPLITTING FOURIER SPECTRAL METHODS FOR THE ALLEN-CAHN EQUATION

  • SHIN, JAEMIN;LEE, HYUN GEUN;LEE, JUNE-YUB
    • Journal of the Korean Society for Industrial and Applied Mathematics
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    • v.21 no.1
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    • pp.1-16
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    • 2017
  • The Allen-Cahn equation is solved numerically by operator splitting Fourier spectral methods. The basic idea of the operator splitting method is to decompose the original problem into sub-equations and compose the approximate solution of the original equation using the solutions of the subproblems. The purpose of this paper is to characterize higher order operator splitting schemes and propose several higher order methods. Unlike the first and the second order methods, each of the heat and the free-energy evolution operators has at least one backward evaluation in higher order methods. We investigate the effect of negative time steps on a general form of third order schemes and suggest three third order methods for better stability and accuracy. Two fourth order methods are also presented. The traveling wave solution and a spinodal decomposition problem are used to demonstrate numerical properties and the order of convergence of the proposed methods.

DIRECT COMPARISON STUDY OF THE CAHN-HILLIARD EQUATION WITH REAL EXPERIMENTAL DATA

  • DARAE, JEONG;SEOKJUN, HAM;JUNSEOK, KIM
    • Journal of the Korean Society for Industrial and Applied Mathematics
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    • v.26 no.4
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    • pp.333-342
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    • 2022
  • In this paper, we perform a direct comparison study of real experimental data for domain rearrangement and the Cahn-Hilliard (CH) equation on the dynamics of morphological evolution. To validate a mathematical model for physical phenomena, we take initial conditions from experimental images by using an image segmentation technique. The image segmentation algorithm is based on the Mumford-Shah functional and the Allen-Cahn (AC) equation. The segmented phase-field profile is similar to the solution of the CH equation, that is, it has hyperbolic tangent profile across interfacial transition region. We use unconditionally stable schemes to solve the governing equations. As a test problem, we take domain rearrangement of lipid bilayers. Numerical results demonstrate that comparison of the evolutions with experimental data is a good benchmark test for validating a mathematical model.

A FAST AND ACCURATE NUMERICAL METHOD FOR MEDICAL IMAGE SEGMENTATION

  • Li, Yibao;Kim, Jun-Seok
    • Journal of the Korean Society for Industrial and Applied Mathematics
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    • v.14 no.4
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    • pp.201-210
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    • 2010
  • We propose a new robust and accurate method for the numerical solution of medical image segmentation. The modified Allen-Cahn equation is used to model the boundaries of the image regions. Its numerical algorithm is based on operator splitting techniques. In the first step of the splitting scheme, we implicitly solve the heat equation with the variable diffusive coefficient and a source term. Then, in the second step, using a closed-form solution for the nonlinear equation, we get an analytic solution. We overcome the time step constraint associated with most numerical implementations of geometric active contours. We demonstrate performance of the proposed image segmentation algorithm on several artificial as well as real image examples.

Underwater Acoustic Lens Design Using Topology Optimization (위상최적화를 이용한 수중음향렌즈의 설계)

  • Jang, Gang-Won;Tran, Quang Dat;Cho, Wan-Ho;Kwon, Hyu-Sang;Cho, Seung Hyun
    • Proceedings of the Korean Society for Noise and Vibration Engineering Conference
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    • 2014.10a
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    • pp.555-556
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    • 2014
  • In this paper, topology optimization of two-dimensional acoustic lenses is presented by using the phase field method. The objective of the optimization is to maximize the acoustic pressure at a specified domain inside the acoustic domain for a given frequency, and the constraint is imposed on the amount of the material of the acoustic lens. Topology optimization of two-dimensional acoustic lenses are obtained as the steady state of the phase transition described by the Allen-Cahn equation. The Helmholtz equation modeling the wave propagation is solved by using a finite element method. The effectiveness of the proposed method is verified by applying it for several two-dimensional acoustic lens system design problems.

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A ROBUST AND ACCURATE PHASE-FIELD SIMULATION OF SNOW CRYSTAL GROWTH

  • Li, Yibao;Lee, Dong-Sun;Lee, Hyun-Geun;Jeong, Da-Rae;Lee, Chae-Young;Yang, Dong-Gyu;Kim, Jun-Seok
    • Journal of the Korean Society for Industrial and Applied Mathematics
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    • v.16 no.1
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    • pp.15-29
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    • 2012
  • In this paper we introduce 6-fold symmetry crystal growth using new phase-field models based on the modified Allen-Cahn equation. The proposed method is a hybrid method which uses both analytic and numerical solutions. We then show this method can be extended to $k$-fold case. The Wulff construction procedure is provided to understand and predict the shape of crystals. We also present a detailed mathematical proof of the validity of the Wulff construction. For computational results, we verify the accuracy and efficiency of the method for snow crystal growth.