A MODIFIED CAHN–HILLIARD EQUATION FOR 3D VOLUME RECONSTRUCTION FROM TWO PLANAR CROSS SECTIONS

SEUNGGYU LEE,YONGHO CHOI,DOYOON LEE,HONG-KWON JO,SEUNGHYUN LEE,SUNGHYUN MYUNG,JUNSEOK KIM 한국산업응용수학회 2015 Journal of the Korean Society for Industrial and A Vol.19 No.1

In this paper, we present an implicit method for reconstructing a 3D solid model from two 2D cross section images. The proposed method is based on the Cahn?Hilliard model for the image inpainting. Image inpainting is the process of reconstructing lost parts of images based on information from neighboring areas. We treat the empty region between the two cross sections as inpainting region and use two cross sections as neighboring information. We initialize the empty region by the linear interpolation. We perform numerical experiments demonstrating that our proposed method can generate a smooth 3D solid model from two cross section data.

COMPARISON OF DIFFERENT NUMERICAL SCHEMES FOR THE CAHN–HILLIARD EQUATION

SEUNGGYU LEE,CHAEYOUNG LEE,HYUN GEUN LEE,JUNSEOK KIM 한국산업응용수학회 2013 Journal of the Korean Society for Industrial and A Vol.17 No.3

The Cahn-Hilliard equation was proposed as a phenomenological model for describing the process of phase separation of a binary alloy. The equation has been applied to many physical applications such as amorphological instability caused by elastic non-equilibrium, image inpainting, two- and three-phase fluid flow, phase separation, flow visualization and the formation of the quantum dots. To solve the Cahn.Hillard equation, many numerical methods have been proposed such as the explicit Euler’s, the implicit Euler’s, the Crank.Nicolson, the semi-implicit Euler’s, the linearly stabilized splitting and the non-linearly stabilized splitting schemes. In this paper, we investigate each scheme in finite-difference schemes by comparing their performances, especially stability and efficiency. Except the explicit Euler’s method, we use the fast solver which is called a multigrid method. Our numerical investigation shows that the linearly stabilized stabilized splitting scheme is not unconditionally gradient stable in time unlike the known result. And the Crank.Nicolson scheme is accurate but unstable in time, whereas the non-linearly stabilized splitting scheme has advantage over other schemes on the time step restriction.

ACCURATE AND EFFICIENT COMPUTATIONS FOR THE GREEKS OF EUROPEAN MULTI-ASSET OPTIONS

SEUNGGYU LEE,YIBAO LI,YONGHO CHOI,HYOUNGSEOK HWANG,JUNSEOK KIM 한국산업응용수학회 2014 Journal of the Korean Society for Industrial and A Vol.18 No.1

This paper presents accurate and efficient numerical methods for calculating the sensitivities of two-asset European options, the Greeks. The Greeks are important financial instruments in management of economic value at risk due to changing market conditions. The option pricing model is based on the Black-Scholes partial differential equation. The model is discretized by using a finite difference method and resulting discrete equations are solved by means of an operator splitting method. For Delta, Gamma, and Theta, we investigate the effect of high-order discretizations. For Rho and Vega, we develop an accurate and robust automatic algorithm for finding an optimal value. A cash-or-nothing option is taken to demonstrate the performance of the proposed algorithm for calculating the Greeks. The results show that the new treatment gives automatic and robust calculations for the Greeks.

Effective Time Step Analysis of a Nonlinear Convex Splitting Scheme for the Cahn-Hilliard Equation

Lee, Seunggyu,Kim, Junseok Global Science Press 2019 Communications in computational physics Vol.25 No.2

The fractional Allen–Cahn equation with the sextic potential

Lee, Seunggyu,Lee, Dongsun Elsevier 2019 Applied mathematics and computation Vol.351 No.-

<P><B>Abstract</B></P> <P>We extend the classical Allen–Cahn (AC) equation to the fractional Allen–Cahn equation (FAC) with triple-well potential. By replacing the spatial Laplacian and double-well potential with fractional Laplacian and triple-well potential, we observe different dynamics. This study leads us to understand different properties of the FAC equation. We seek the existence, boundedness, and unique solvability of numerical solutions for the FAC equation with triple-well potential. In addition, the inclusion principle for the Allen–Cahn equation is considered. These different properties make us enhance the applicability of the phase-field method to the mathematical modeling in materials science. In computation, the spectral decomposition for the fractional operator allows us to develop a numerical method for the fractional Laplacian problem. For the periodic and discrete Laplacian matrix and vector multiplication, circulant submatices are formed in more than two-dimensional case. Even if the fast Fourier transform (FFT) can be utilized in this modeling, we construct the inverse of the doubly-block-circulant matrix for the solution of the fractional Allen–Cahn equation. In doing so, it helps to straightforwardly understand the numerical treatment, and exploit the properties of the discrete Fourier transforms.</P> <P><B>Highlights</B></P> <P> <UL> <LI> Boundedness of numerical solutions for the fractional Allen – Cahn equation with sextic polynomial is under consideration. </LI> <LI> From the numerical aspect, the inclusion principle is proven to hold only for the Allen – Cahn equation. </LI> <LI> Effects of the fractional order and parameters of the equation is analyzed. </LI> <LI> Eigenvectors of the discrete Laplacian are used for investigating the behaviors of the fractional Allen – Cahn equation. </LI> <LI> The numerical study of the fractional Allen – Cahn equation with the sextic polynomial is verified by various simulations. </LI> </UL> </P>

Phase-field simulations of crystal growth in a two-dimensional cavity flow

Lee, Seunggyu,Li, Yibao,Shin, Jaemin,Kim, Junseok Elsevier 2017 Computer physics communications Vol.216 No.-

<P><B>Abstract</B></P> <P>In this paper, we consider a phase-field model for dendritic growth in a two-dimensional cavity flow and propose a computationally efficient numerical method for solving the model. The crystal is fixed in the space and cannot be convected in most of the previous studies, instead the supercooled melt flows around the crystal, which is hard to be realized in the real world experimental setting. Applying advection to the crystal equation, we have problems such as deformation of crystal shape and ambiguity of the crystal orientation for the anisotropy. To resolve these difficulties, we present a phase-field method by using a moving overset grid for the dendritic growth in a cavity flow. Numerical results show that the proposed method can predict the crystal growth under flow.</P>

COMPARISON OF NUMERICAL METHODS FOR TERNARY FLUID FLOWS

SEUNGGYU LEE,DARAE JEONG,YONGHO CHOI,JUNSEOK KIM 한국산업응용수학회 2016 Journal of the Korean Society for Industrial and A Vol.20 No.1

This paper reviews and compares three different methods for modeling incompressible and immiscible ternary fluid flows: the immersed boundary, level set, and phase-field methods. The immersed boundary method represents the moving interface by tracking the Lagrangian particles. In the level set method, an interface is defined implicitly by using the signed distance function, and its evolution is governed by a transport equation. In the phase-field method, the advective Cahn–Hilliard equation is used as the evolution equation, and its order parameter also implicitly defines an interface. Each method has its merits and demerits. We perform the several simulations under different conditions to examine the merits and demerits of each method. Based on the results, we determine the most suitable method depending on the specific modeling needs of different situations.

Mathematical modelling for investigating the function of anatomical structures

Seunggyu Lee,Mi-Sun Hur,Chang-Seok Oh 대한체질인류학회 2021 대한체질인류학회 학술대회 연제 초록 Vol.64 No.-

Mathematics is considered as a useful scientific tool in various fields such as multiphase fluid dynamics, image processing, dendrite growth, and cell dynamics. Since Allen Turing (1952) proposed a concept of the reaction-diffusion equation for pattern formation, the mathematical and computational biology has been actively studied in many areas of both basic and clinical medicines. In this talk, we discuss some mathematical models used in our recent studies of anatomical structures and their corresponding numerical simulation results. It is presented how mathematics and 21st century anatomy meet to investigate the function of human organs; the longitudinal muscular column in the prostatic urethral wall (Prostate 80(6), 2020), the facial muscles acting on the nasolabial fold (PLoS One 15(8), 2020), the channels in the interatrial septum of the heart (PLoS One 16(2), 2021), and the structure and function of pyloric sphincter in preparation. The immersed boundary and phase-field methods, which are coupled systems of the Navier-Stokes and transport equations have been mainly applied for the research.

COMPARISON OF DIFFERENT NUMERICAL SCHEMES FOR THE CAHN-HILLIARD EQUATION

Seunggyu Lee,Jaemin Shin,Darae Jeong,Hyun Geun Lee,Junseok Kim 한국산업응용수학회 2011 한국산업응용수학회 학술대회 논문집 Vol.6 No.1

In this work, we numerically analyze a class of time discretizations for the Cahn-Hilliard equation. It is useful to investigate the performance of different schemes in terms of accuracy and efficiency since these schemes are frequently used in various science applications. In this work, comparisons of the explicit Euler’s, implicit Euler’s, Crank-Nicolson, semi-implicit Euler’s, linearly stabilized splitting, and non-linearly stabilized splitting schemes are presented. The continuous problem has the conservation of mass and the decrease of the total energy. We check the same properties hold for the discrete problem.

Experimental and numerical investigations of near-field underwater explosions

Seunggyu Lee,Junghee Cho,Chaemin Lee,Seongpil Cho 국제구조공학회 2021 Structural Engineering and Mechanics, An Int'l Jou Vol.77 No.3

Near-field underwater explosion (UNDEX) phenomena were investigated by experiments and numerical simulations. The UNDEX experiments were performed in a water tank using a ship-like model. One kilogram of TNT, one of the most widely used military high explosives, was used for the experiments. Numerical simulations were performed under the same conditions as in the experiments using the commercial software LS-DYNA. Underwater pressures, accelerations, velocities, and strains by shock waves were measured at multiple locations. Further, the bubble pulsation period and the whipping deformations of the ship-like model were explored. The experimental results are presented and examined through comparison with the results obtained from widely used empirical equations and numerical simulations.