Hyun Geun Lee

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.

A semi-analytical Fourier spectral method for the Allen-Cahn equation

In recent years, Fourier spectral methods have been widely used as a powerful tool for solving phase-field equations. To improve its effectiveness, many researchers have employed stabilized semi-implicit Fourier spectral (SIFS) methods which allow a much larger time step than a usual explicit scheme. Our mathematical analysis and numerical experiments, however, suggest that an effective time step is smaller than a time step specified in the SIFS schemes. In consequence, the SIFS scheme is inaccurate for a considerably large time step and may lead to incorrect morphologies in phase separation processes. In order to remove the time step constraint and guarantee the accuracy in time for a sufficiently large time step, we present a first and a second order semi-analytical Fourier spectral (SAFS) methods for solving the Allen-Cahn equation. The core idea of the methods is to decompose the original equation into linear and nonlinear subequations, which have closed-form solutions in the Fourier and physical spaces, respectively. Both the first and the second order methods are unconditionally stable and numerical experiments demonstrate that our proposed methods are more accurate than the stabilized semi-implicit Fourier spectral method.

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.

First and second order operator splitting methods for the phase field crystal equation

In this paper, we present operator splitting methods for solving the phase field crystal equation which is a model for the microstructural evolution of two-phase systems on atomic length and diffusive time scales. A core idea of the methods is to decompose the original equation into linear and nonlinear subequations, in which the linear subequation has a closed-form solution in the Fourier space. We apply a nonlinear Newton-type iterative method to solve the nonlinear subequation at the implicit time level and thus a considerably large time step can be used. By combining these subequations, we achieve the first- and second-order accuracy in time. We present numerical experiments to show the accuracy and efficiency of the proposed methods.

An efficient and accurate numerical algorithm for the vector-valued Allen–Cahn equations

a b s t r a c t In this paper, we consider the vector-valued Allen–Cahn equations which model phase separation in N-component systems. The considerations of solving numerically the vector-valued Allen–Cahn equations are as follows: (1) the use of a small time step is appropriate to obtain a stable solution and (2) a sufficient number of phase-field variables is required to capture the correct dynamics. However, stability restrictions on the time step and a large number of phase-field variables cause huge computational costs and make the calculation very inefficient. To overcome this problem, we present an efficient and accurate numerical algorithm which is based on an operator splitting technique and is solved by a fast solver such as a linear geometric multigrid method. The algorithm allows us to convert the vector-valued Allen–Cahn equations with N components into a system of N − 1 binary Allen–Cahn equations and drastically reduces the required computational time and memory. We demonstrate the efficiency and accuracy of the algorithm with various numerical experiments. Furthermore, using our algorithm, we can simulate the growth of multiple crystals with different orientation angles and fold numbers on a single domain. Finally, the efficiency of our algorithm is validated with an example that includes the growth of multiple

Numerical simulation of the three-dimensional Rayleigh-Taylor instability

The Rayleigh-Taylor instability is a fundamental instability of an interface between two fluids of different densities, which occurs when the light fluid is pushing the heavy fluid. During the nonlinear stages, the growth of the Rayleigh-Taylor instability is greatly affected by three-dimensional effects. To investigate three-dimensional effects on the Rayleigh-Taylor instability, we introduce a new method of computation of the flow of two incompressible and immiscible fluids and implement a time-dependent pressure boundary condition that relates to a time-dependent density field at the domain boundary. Through numerical examples, we observe the two-layer roll-up phenomenon of the heavy fluid, which does not occur in the two-dimensional case. And by studying the positions of the bubble front, spike tip, and saddle point, we show that the three-dimensional Rayleigh-Taylor instability exhibits a stronger dependence on the density ratio than on the Reynolds number. Finally, we perform a long time three-dimensional simulation resulting in an equilibrium state.

Choindroitinase ABC I-Mediated Enhancement of Oncolytic Virus Spread and Anti Tumor Efficacy: A Mathematical Model

Oncolytic viruses are genetically engineered viruses that are designed to kill cancer cells while doing minimal damage to normal healthy tissue. After being injected into a tumor, they infect cancer cells, multiply inside them, and when a cancer cell is killed they move on to spread and infect other cancer cells. Chondroitinase ABC (Chase-ABC) is a bacterial enzyme that can remove a major glioma ECM component, chondroitin sulfate glycosoamino glycans from proteoglycans without any deleterious effects in vivo. It has been shown that Chase-ABC treatment is able to promote the spread of the viruses, increasing the efficacy of the viral treatment. In this paper we develop a mathematical model to investigate the effect of the Chase-ABC on the treatment of glioma by oncolytic viruses (OV). We show that the models predictions agree with experimental results for a spherical glioma. We then use the model to test various treatment options in the heterogeneous microenvironment of the brain. The model predicts that separate injections of OV, one into the center of the tumor and another outside the tumor will result in better outcome than if the total injection is outside the tumor. In particular, the injection of the ECM-degrading enzyme (Chase-ABC) on the periphery of the main tumor core need to be administered in an optimal strategy in order to infect and eradicate the infiltrating glioma cells outside the tumor core in addition to proliferative cells in the bulk of tumor core. The model also predicts that the size of tumor satellites and distance between the primary tumor and multifocal/satellite lesions may be an important factor for the efficacy of the viral therapy with Chase treatment.

Finite Element Analysis of Schwarz P Surface Pore Geometries for Tissue-Engineered Scaffolds

Tissue engineering scaffolds provide temporary mechanical support for tissue regeneration. To regenerate tissues more efficiently, an ideal structure of scaffolds should have appropriate porosity and pore structure. In this paper, we generate the Schwarz primitive (P) surface with various volume fractions using a phase-field model. The phase-field model enables us to design various surface-to-volume ratio structures with high porosity and mechanical properties. Comparing the Schwarz P surfaces von Mises stress with that of triply periodic cylinders and cubes, we draw conclusions about the mechanical properties of the Schwarz P surface.

First and second order numerical methods based on a new convex splitting for phase-field crystal equation

The phase-field crystal equation derived from the SwiftHohenberg energy functional is a sixth order nonlinear equation. We propose numerical methods based on a new convex splitting for the phase-field crystal equation. The first order convex splitting method based on the proposed splitting is unconditionally gradient stable, which means that the discrete energy is non-increasing for any time step. The second order scheme is unconditionally weakly energy stable, which means that the discrete energy is bounded by its initial value for any time step. We prove mass conservation, unique solvability, energy stability, and the order of truncation error for the proposed methods. Numerical experiments are presented to show the accuracy and stability of the proposed splitting methods compared to the existing other splitting methods. Numerical tests indicate that the proposed convex splitting is a good choice for numerical methods of the phase-field crystal equation.

High-order and mass conservative methods for the conservative Allen-Cahn equation

The conservative Allen-Cahn (AC) equation has been studied analytically and numerically. Our mathematical analysis and numerical experiment, however, show that previous numerical methods are not second-order accurate in time and/or do not conserve the initial mass. The aim of this paper is to propose high-order and mass conservative methods for solving the conservative AC equation. In the methods, we discretize the conservative AC equation by using a Fourier spectral method in space and first-, second-, and third-order implicit-explicit Runge-Kutta schemes in time. We show that the methods inherit the mass conservation. Numerical experiments are presented demonstrating the accuracy and efficiency of proposed methods.