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Li, Yibao,Luo, Chaojun,Xia, Binhu,Kim, Junseok Butterworths [etc.] 2019 Applied mathematical modelling Vol.67 No.-
<P><B>Abstract</B></P> <P>We develop an unconditionally stable direct discretization scheme for solving the phase-field crystal equation on surfaces. The surface is discretized by using an unstructured triangular mesh. Gradient, divergence, and Laplacian operators are defined on triangular meshes. The proposed numerical method is second-order accurate in space and time. At each time step, the proposed computational scheme results in linear elliptic equations to be solved, thus it is easy to implement the algorithm. It is proved that the proposed scheme satisfies a discrete energy-dissipation law. Therefore, it is unconditionally stable. A fast and efficient biconjugate gradients stabilized solver is used to solve the resulting discrete system. Numerical experiments are conducted to demonstrate the performance of the proposed algorithm.</P> <P><B>Highlights</B></P> <P> <UL> <LI> An efficient method for phase-field crystal equation on surfaces is proposed. </LI> <LI> The proposed scheme is second-order accurate in time and space. </LI> <LI> The unconditional stability of the proposed scheme is analytically proved. </LI> <LI> The resulting system of discrete equations is easy to implement. </LI> </UL> </P>
Li, Yibao,Kim, Junseok North-Holland Pub. Co 2017 Computer methods in applied mechanics and engineer Vol.319 No.-
<P><B>Abstract</B></P> <P>In this paper, we present a high-order accurate compact scheme for the phase field crystal model in two- and three-dimensional spaces. The proposed scheme is derived by combining a fourth-order compact finite difference formula in space and a backward differentiation for the time derivative term, which is second-order accurate in time. Furthermore, a nonlinearly stabilized splitting scheme is used and thus a larger time step can be allowed. Since the equations at the implicit time level are nonlinear, we introduce a Newton-type iterative method and employ a fast and efficient nonlinear multigrid solver to solve the resulting discrete system. In particular, we implement the compact scheme in the adaptive mesh refinement framework. An adaptive time step method for the phase field crystal model is also proposed. Various numerical experiments are presented and confirm the accuracy, stability, and efficiency of our proposed method.</P> <P><B>Highlights</B></P> <P> <UL> <LI> We propose an efficient and stable compact fourth-order method for the phase field crystal equation. </LI> <LI> The proposed numerical method with second order accuracy of time exhibits excellent stability. </LI> <LI> We implement the compact scheme in the adaptive mesh refinement framework. </LI> <LI> An adaptive time step method for the phase field crystal model is also presented. </LI> </UL> </P>
Surface reconstruction from unorganized points with <i>l</i> <sub>0</sub> gradient minimization
Li, Huibin,Li, Yibao,Yu, Ruixuan,Sun, Jian,Kim, Junseok Elsevier 2018 Computer vision and image understanding Vol.169 No.-
<P><B>Abstract</B></P> <P>To reconstruct surface from unorganized points in three-dimensional Euclidean space, we propose a novel efficient and fast method by using <I>l</I> <SUB>0</SUB> gradient minimization, which can directly measure the sparsity of a solution and produce sharper surfaces. Therefore, the proposed method is particularly effective for sharpening major edges and removing noise. Unlike the Poisson surface reconstruction approach and its extensions, our method does not depend on the accurate directions of normal vectors of the unorganized points. The resulting algorithm is developed using a half-quadratic splitting method and is based on decoupled iterations that are alternating over a smoothing step realized by a Poisson approach and an edge-preserving step through an optimization formulation. This iterative algorithm is easy to implement. Various tests are presented to demonstrate that our method is robust to point noise, normal noise and data holes, and thus produces good surface reconstruction results.</P> <P><B>Highlights</B></P> <P> <UL> <LI> With l0 gradient minimization, a fast and efficient surface reconstruction method is proposed. </LI> <LI> Our method is particularly effective for sharpening major edges and removing noises. </LI> <LI> Our method does not depend on the accuracy of normal vectors of the unorganized points. </LI> </UL> </P>
AN UNCONDITIONALLY STABLE HYBRID NUMERICALMETHOD FOR THE ALLEN-CAHN EQUATION
Yibao LI,Hyun Geun LEE,Darae JEONG,Junseok KIM 한국산업응용수학회 2009 한국산업응용수학회 학술대회 논문집 Vol.4 No.2
We present an unconditionally stable second-order hybrid numerical method for solving the Allen-Cahn equation representing a model for antiphase domain coarsening in a binary mixture. The proposed method is based on operator splitting techniques. The Allen-Cahn equation was divided into a linear and a nonlinear equation. First, the linear equation was solved using a Crank-Nicolson scheme, and a fast solver, such as a multigrid method, was applied. The nonlinear equation was then solved analytically due to the availability of a closed-form solution. Various numerical experiments are presented to confirm the accuracy, efficiency, and stability of the proposed method. In particular, we show that the scheme is unconditionally stable and second order accurate in both time and space.
A fast, robust, and accurate operator splitting method for the crystal growth simulation
Yibao Li,Junseok Kim 한국산업응용수학회 2009 한국산업응용수학회 학술대회 논문집 Vol.4 No.2
In this paper, a fast, robust, and accurate operator splitting method for the crystal growth simulation of binary alloy solidification both in 2D and 3D is present. The proposed method is based on operator splitting techniques and well described by four numerical solutions. We present numerical experiments for crystal simulation to demonstrate the accuracy and efficiency of the proposed method.
Phase-field simulations of crystal growth with adaptive mesh refinement
Yibao Li,Junseok Kim 한국산업응용수학회 2011 한국산업응용수학회 학술대회 논문집 Vol.6 No.1
In this paper, we propose the phase-field simulation of dendritic growth in both two- and three-dimensional space with adaptive mesh refinement. The proposed method is based on operator splitting techniques which we have published in Journal of Crystal Growth. We also present a set of representative numerical experiments for crystal growth simulation to demonstrate the accuracy of the proposed method. Our simulation results are also consistent with previous numerical experiments.
Adaptive mesh refinement computations of crystal growth using a phase-field model
Yibao Li,Junseok Kim 한국산업응용수학회 2012 한국산업응용수학회 학술대회 논문집 Vol.7 No.2
In this paper, we propose the phase-field simulation of dendritic crystal growth in both two- and three-dimensional spaces with adaptive mesh refinement, which was designed to solve nonlinear parabolic partial differential equations. The proposed numerical method is based on operator splitting techniques and extends the previous research [Y. Li, H.G. Lee, J. Kim, A fast, robust, and accurate operator splitting method for phase-field simulations of crystal growth, J. Cryst. Growth 321 (2011) 176?182.]. The resulting discrete system of equations is solved by a fast numerical method such as an adaptive multigrid method. Comparisons to uniform mesh method, explicit adaptive method, and previous numerical experiments for crystal growth simulations are presented to demonstrate the accuracy and robustness of the proposed method.
A FAST AND ACCURATE NUMERICAL METHOD FOR MEDICAL IMAGE SEGMENTATION
YIBAO LI,JUNSEOK KIM 한국산업응용수학회 2010 Journal of the Korean Society for Industrial and A Vol.14 No.4
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.
Yibao Li,Darae Jeong,Jaemin Shin,Junseok Kim 한국산업응용수학회 2012 한국산업응용수학회 학술대회 논문집 Vol.7 No.1
In this paper we present a conservative numerical method for the Cahn?Hilliard equation with Dirichlet boundary conditions in complex domains. The method uses an unconditionally gradient stable nonlinear splitting numerical scheme to remove the high-order time-step stability constraints. The continuous problem has conservation of mass and we prove the conservative property of the proposed discrete scheme for complex domains. The resulting system of discrete equations is solved by a nonlinear multigrid method. We describe the implementation of the proposed numerical scheme in more detail. We demonstrate the accuracy and robustness of the proposed Dirichlet boundary formulation using various numerical experiments. We numerically show the total energy decrease and the unconditionally gradient stability. In particular, the numerical results indicate the potential usefulness of the proposed method for accurately calculating biological membrane dynamics in confined domains.