Junseok Kim

Quantum dot formation on a strain-patterned epitaxial thin film

We model the effect of substrate strain patterning on the self-assembly of quantum dots (QDs). When the surface energy is isotropic, we demonstrate that strain patterning via embedded substrate inclusions may result in ordered, self-organized QD arrays. However, for systems with strong cubic surface energy anisotropy, the same patterning does not readily lead to an ordered array of pyramids at long times. We conclude that the form of the surface energy anisotropy strongly influences the manner in which QDs self-assemble into regular arrays.

Solving the regularized, strongly anisotropic Cahn–Hilliard equation by an adaptive nonlinear multigrid method

Abstract We present efficient, second-order accurate and adaptive finite-difference methods to solve the regularized, strongly anisotropic Cahn–Hilliard equation in 2D and 3D. When the surface energy anisotropy is sufficiently strong, there are missing orientations in the equilibrium level curves of the diffuse interface solutions, corresponding to those missing from the sharp interface Wulff shape, and the anisotropic Cahn–Hilliard equation becomes ill-posed. To regularize the equation, a higher-order derivative term is added to the energy. This leads to a sixth-order, nonlinear parabolic equation for the order parameter. An implicit time discretization is used to remove the high-order time step stability constraints. Dynamic block-structured Cartesian mesh refinement is used to highly resolve narrow interfacial layers. A multilevel, nonlinear multigrid method is used to solve the nonlinear equations at the implicit time level. One of the keys to the success of the method is the treatment of the anisotropic term. This term is discretized in conservation form in space and is discretized fully implicitly in time. Numerical simulations are presented that confirm the accuracy, efficiency and stability of the scheme. We study the dynamics of interfaces under strong anisotropy and compare near-equilibrium diffuse interface solutions to the sharp interface Wulff shapes in 2D and 3D. We also simulate large-scale coarsening of a corrugated surface (in 3D) evolving by anisotropic surface diffusion. We show the emergence of long-range order during coarsening and an interesting mechanism of ordered coarsening.

Phase field modeling and simulation of three-phase flows

Abstract. In this paper, we derive a thermodynamically consistent pha se-field model for flows containing three (or more) liquid components. The model is based on a Navier-S tokes (NS) and Cahn-Hilliard system (CH) which accounts for surface tension among the different component s a d three-phase contact lines. We develop a stable conservative, second order accurate fully implicit discre tization of the NS and three-phase (ternary) CH system. We use a nonlinear multigrid method to efficiently solve the dis crete ternary CH system at the implicit time-level and then couple it to a multigrid/projection method that is used to solve the NS equation. We demonstrate convergence of our scheme numerically and perform numerical simulation s to show the accuracy, flexibility, and robustness of this approach. In particular, we simulate a three interface contact angle resulting from a spreading liquid lens on an interface, a buoyancy-driven compound drop, and the Rayl eigh-Taylor instability of a flow with three partially miscible components.

An unconditionally stable hybrid numerical method for solving the Allen-Cahn equation

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 discretized using a Crank-Nicolson scheme and the resulting discrete system of equations was solved by a fast solver such as a multigrid method. 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.

Multiphase image segmentation using a phase-field model

In this paper, we propose a new, fast, and stable hybrid numerical method for multiphase image segmentation using a phase-field model. The proposed model is based on the Allen-Cahn equation with a multiple well potential and a data-fitting term. The model is computationally superior to the previous multiphase image segmentation via Modica-Mortola phase transition and a fitting term. We split its numerical solution algorithm into linear and a nonlinear equations. The linear equation is discretized using an implicit scheme and the resulting discrete system of equations is solved by a fast numerical method such as a multigrid method. The nonlinear equation is solved analytically due to the availability of a closed-form solution. We also propose an initialization algorithm based on the target objects for the fast image segmentation. Finally, various numerical experiments on real and synthetic images with noises are presented to demonstrate the efficiency and robustness of the proposed model and the numerical method.

A conservative numerical method for the Cahn-Hilliard equation in complex domains

We propose an efficient finite difference scheme for solving the Cahn-Hilliard equation with a variable mobility in complex domains. Our method employs a type of unconditionally gradient stable splitting discretization. We also extend the scheme to compute the Cahn-Hilliard equation in arbitrarily shaped domains. We prove the mass conservation property of the proposed discrete scheme for complex domains. The resulting discretized equations are solved using a multigrid method. Numerical simulations are presented to demonstrate that the proposed scheme can deal with complex geometries robustly. Furthermore, the multigrid efficiency is retained even if the embedded domain is present.

A comparison study of ADI and operator splitting methods on option pricing models

In this paper we perform a comparison study of alternating direction implicit (ADI) and operator splitting (OS) methods on multi-dimensional Black-Scholes option pricing models. The ADI method is used extensively in mathematical finance for numerically solving multi-factor option pricing problems. However, numerical results from the ADI scheme show oscillatory solution behaviors with nonsmooth payoffs or discontinuous derivatives at the exercise price with large time steps. In the ADI scheme, there are source terms which include y-derivatives when we solve x-derivative involving equations. Then, due to the nonsmooth payoffs, source terms contain abrupt changes which are not in the range of implicit discrete operators and this leads to difficulty in solving the problem. On the other hand, the OS method does not contain the other variables derivatives in the source terms. We provide computational results showing the performance of the methods for two-asset option pricing problems. The results show that the OS method is very efficient and gives better accuracy and robustness than the ADI method with large time steps.

A phase-field approach for minimizing the area of triply periodic surfaces with volume constraint

In this paper, we present an accurate and efficient algorithm to generate constant mean curvature surfaces with volume constraint using a phase-field model. We implement our proposed algorithm using an unconditionally gradient stable nonlinear splitting scheme. Starting from the periodic nodal surface approximation to minimal surfaces, we can generate various constant mean curvature surfaces with given volume fractions. We generate and study the Schwarz primitive (P), Schwarz diamond (D), and Schoen gyroid (G) surfaces with various volume fractions. This technique for generating constant mean curvature surfaces can be used to design biomedical scaffolds with optimal mechanical and biomorphic properties.

A conservative numerical method for the Cahn-Hilliard equation with Dirichlet boundary conditions in complex domains

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 the conservation of mass and we prove the conservative property of the proposed discrete scheme in complex domains. We describe the implementation of the proposed numerical scheme in detail. The resulting system of discrete equations is solved by a nonlinear multigrid method. 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.

An unconditionally stable numerical method for bimodal image segmentation

In this paper, we propose a new level set-based model and an unconditionally stable numerical method for bimodal image segmentation. Our model is based on the Lee–Seo active contour model. The numerical scheme is semi-implicit and solved by an analytical method. The unconditional stability of the proposed numerical method is proved analytically. We demonstrate performance of the proposed image segmentation algorithm on several synthetic and real images to confirm the efficiency and stability of the proposed method.