Semi-implicit Fourier Spectral Method Research Articles

Performance Benchmark of Cahn–Hilliard Equation Solver with Implementation of Semi-implicit Fourier Spectral Method

Performance Benchmark of Cahn–Hilliard Equation Solver with Implementation of Semi-implicit Fourier Spectral Method

A topology constrained phase field model

This paper presents a topology constrained phase field model based on a variable mobility Cahn-Hilliard equation. Mostly in the level set framework, this issue has been addressed using concepts from digital topology. However, point based specifications can not be effective in phase fields where the interface is represented by a diffuse region. In phase field models, topology constraints are included in the energy as a penalty term. However, there are no studies which provide topological constraints in the Cahn-Hilliard model to our knowledge. In this work, we use the mobility parameter to enforce topology constraints; this allows local, explicit control of topological events, unlike the optimization based solutions. We define topologically critical regions in phase fields based on a topological analysis of phase field evolution. This analysis shows that diffuse layers formed at topological encounters behave like a Morse function and the critical points of these transitional layers give topologically critical points in phase fields. Using this property, we first detect critical points using an efficient image analysis method. Then we detect the transitional region (critical region) around each critical point using a local morphological segmentation algorithm. We define a mobility function which both has a spatial and concentration dependence. This function is designed to reverse the effect of diffuse layers formed at topological encounters and prevent the further development of topological events. Being extracted from phase field values, the mobility map naturally aligns with phase field evolution. Together with the use of an unconditionally stable semi-implicit Fourier spectral method for the variable mobility Cahn-Hilliard equation, an efficient model is provided. Local, explicit control mechanism allows the design of free energy functional for modeling different materials; this extends the use of Cahn-Hilliard equation to applications which require topology control.

Numerical solution to phase-field model of solidification: A review

Recent advances in improving the computational efficiency of the phase-field simulations of solidification microstructures are reviewed. The parallel progress of four typical approaches, namely, multigrid, adaptive mesh refinement method, semi-implicit Fourier spectral method, and graphical processing units (GPUs) architecture, is highlighted. Large-scale spatiotemporal simulations are successfully performed to cover essential aspects of multiphysics with the capability of these algorithms. Focus is put on the principles, applications, and comparison of the four algorithms, while solidification theories and discretization methods are outlined.

A study of morphology of silicon with femtosecond laser based on phase field model

In this paper, phase field model is employed to simulate the process of femtosecond laser which irradiates on silicon surface. The numerical model is mainly used to study the micro/nano evolution process and the atomic diffusion motion of the system. Meanwhile, the Semi-Implicit Fourier spectral method and Semi-Implicit Backwards Differentiation Formula are applied for high efficiency and numerical stability. Further, the rules of surface bumps are controlled by different thermal conductivities which are systematically revealed. Our work provides a new prospect for fabricating the bumps of the probe, to obtain high sensitivity sensors in chemical and biological fields.

Novel patterns in a class of fractional reaction–diffusion models with the Riesz fractional derivative

Pattern is a non-uniform macro structure with some regularity in space or time, which is common in nature. In this manuscript, we introduce the Fourier transform for spatial discretization and Runge–Kutta method for time discretization to solve a class of fractional reaction–diffusion models such as Allen–Cahn model, FitzHugh–Nagumo model and Gray–Scott model with space derivatives described by the fractional Laplacian. Numerical experiments show that compared with semi-implicit Fourier spectral method, present method has higher precision and low computational complexity. The patterns of 2D FitzHugh–Nagumo model with standard diffusion obtained in this manuscript are in line with the numerical simulations and theoretical analysis that made by other academics. Then, we discuss the limit case of fractional order: the process of pattern formations of the fractional order tends to corresponding integer-order reaction–diffusion model when super diffusion tends to standard diffusion. Finally, some long time diffusion behaviors of 3D FitzHugh–Nagumo model and 3D Gray–Scott model are observed.

Effect of Al Concentration on the Microstructural Evolution of Fe-Cr-Al Systems: A Phase-Field Approach

In this study, the microstructural evolution of an Fe-Cr-Al system was simulated in two-dimensional (2D) and three-dimensional (3D) systems using the phase-field method. We investigated the effect of Al concentration on the microstructural evolution of the systems, with a focus on the nucleation and growth of the Cr-rich α′ phase. In addition, we quantitatively analyzed the mechanism of the effect of Al concentration on the microstructural characteristics of the 2D and 3D systems, such as the number of precipitates, average precipitate area (volume), and α′ phase fraction. In particular, we analyzed the effect of Al concentration and the dimensions of the system cell on the formation of the interconnected structure at high Cr concentrations, such as 35 Crat% and 40 Crat%. To enhance the performance of the simulations, we applied a semi-implicit Fourier spectral method for the ternary system and a parallel graphics processing unit computing technique. The results revealed that the initiation of phase separation in the 2D and 3D simulations was enhanced with an increase in the average Al concentration in the system. In addition, with an increase in the average Al concentration in both systems, the α′ phase fraction increased, while the change in the phase fraction decreased.

Phase Field Modelling of Precipitate Coarsening in Binary Alloys with Respect to Atomic Mobility of Solute in the Precipitate Phase

In the current work, a phase field model is used to study the effect of atomic mobility inside the precipitate phase on coarsening behaviour in two-dimensional (2D) systems. In all the coarsening theories until now, the diffusivity inside the precipitate phase has not been explicitly considered which would imply that there is no chemical potential gradient inside the precipitate. The aim of this study is to assess the potential effect of diffusivity inside the precipitate on coarsening behaviour by examining systems with moderate volume fractions, i.e. $$f_{\mathrm{v}}=0.2$$ , $$f_{\mathrm{v}}=0.3$$ and variable atomic mobilities inside the precipitate phase. The atomic mobility inside the matrix is considered to be constant. The Cahn–Hilliard (C–H)-type nonlinear diffusion equation with variable mobility is numerically discretized and solved for a two-phase coarsening system, using a semi-implicit Fourier spectral method. Coarsening behaviour is studied and analysed by keeping in track, the variation of the number density of precipitates, their average size and size distribution with respect to time. The key observation is that, the atomic mobility inside the precipitate phase has a substantial effect on the coarsening kinetics of precipitates. The effect is more clearly observed when the atomic mobility inside the precipitate phase is greater than that of matrix.

A stabilized semi-implicit Fourier spectral method for nonlinear space-fractional reaction-diffusion equations

The reaction-diffusion model can generate a wide variety of spatial patterns, which has been widely applied in chemistry, biology, and physics, even used to explain self-regulated pattern formation in the developing animal embryo. In this work, a second-order stabilized semi-implicit time-stepping Fourier spectral method for the reaction-diffusion systems of equations with space described by the fractional Laplacian is developed. We adopt the temporal-spatial error splitting argument to illustrate that the proposed method is stable without imposing the CFL condition, and an optimal L2-error estimate in space is proved. We also analyze the linear stability of the stabilized semi-implicit method and obtain a practical criterion to choose the time step size to guarantee the stability of the semi-implicit method. Our approach is illustrated by solving several problems of practical interest, including the fractional Allen-Cahn, Gray-Scott and FitzHugh-Nagumo models, together with an analysis of the properties of these systems in terms of the fractional power of the underlying Laplacian operator, which are quite different from the patterns of the corresponding integer-order model.

Effect of magnetic ordering on the spinodal decomposition of the Fe-Cr system: A GPU-accelerated phase-field study

This study aims to investigate the effect of magnetic ordering on spinodal decomposition behavior in the Fe-Cr system using graphics processing unit(GPU) parallelization. We modify the CALPHAD-type free energy to remove its critical behavior near the paramagnetic-ferromagnetic transition. We solve the Cahn-Hilliard equation using a semi-implicit Fourier spectral method, parallelizing the code to run on a GPU via compute unified device architecture and OpenMP. We find that the GPU parallelization gives better performance than that of OpenMP when using fast Fourier transforms to solve the Cahn-Hilliard equation. We conduct nine sets of simulations to examine the effect of magnetic ordering, and we found that it alters the interfacial energy between Cr-rich and Cr-depleted phases, equilibrium concentrations, and energy barrier for phase transformations. We apply a phase-field method to examine in detail how these changes affect the microstructural evolution, quantitatively evaluating the microstructures obtained in terms of the precipitate number density, average phase area, and phase boundary density along certain auxiliary lines to analyze the effects of magnetic ordering.

Phase separation dynamics in binary systems containing mobile particles with variable Brownian motion

This study investigates the spinodal decomposition dynamics in binary mixtures containing mobile particles by combining the Cahn–Hilliard equation with Langevin dynamics for particles with Brownian motion changes proportional to their mobility. We solve the Cahn–Hilliard equation numerically using a semi-implicit Fourier spectral method, and show that the domain growth rate first increases with the increase in particle mobility, and then decreases. The effect of filler particle concentration on the domain growth depends on its mobility: when the particle mobility is low, the domain growth rate decreases with the increase in particle concentration; whereas when the particle mobility is high, the domain growth rate decreases and then increases and finally decreases again with the increase in particle concentration. The proposed model suggests the possibility of controlling macroscopic behaviour of binary alloys by altering filler particle properties.

Fourier Spectral Phase Field Simulations of Elastically Anisotropic Heterogeneous Polycrystals

Abstract In this study, we developed a multi-order, phase field model to compute the stress distributions in anisotropically elastic, inhomogeneous polycrystals and study stress-driven grain boundary migration. In particular, we included elastic contributions to the total free energy density and solved the multicomponent, nonconserved Allen–Cahn equations via the semi-implicit Fourier spectral method. Our analysis included specific cases related to bicrystalline planar and curved systems as well as polycrystalline systems with grain orientation and applied strain conditions. The evolution of the grain boundary confirmed the strong dependencies between grain orientation and applied strain conditions and the localized stresses were found to be maximum within grain boundary triple junctions.

Simulation of the first order phase transitions in binary alloys with variable mobility

The first order phase transitions in binary alloys were simulated basing on the Cahn–Hilliard equation for metastable states with mobility depending on the local composition. The simulation was carried out utilizing the semi-implicit Fourier spectral method for 3D fragment of a solid solution satisfying the regular solution approximation. We defined kinetics of the main characteristics of phase distribution: nucleation rate, average size, concentration of precipitates and autocorrelation function etc. Peculiarities of different stages of binary alloy decomposition (nucleation, diffusion growth and coarsening) were analyzed both for constant and variable mobility.

Characterizing the Stabilization Size for Semi-Implicit Fourier-Spectral Method to Phase Field Equations

Recent results in the literature provide computational evidence that stabilized semi-implicit time-stepping method can efficiently simulate phase field problems involving fourth-order nonlinear dif- fusion, with typical examples like the Cahn-Hilliard equation and the thin film type equation. The up-to-date theoretical explanation of the numerical stability relies on the assumption that the deriva- tive of the nonlinear potential function satisfies a Lipschitz type condition, which in a rigorous sense, implies the boundedness of the numerical solution. In this work we remove the Lipschitz assumption on the nonlinearity and prove unconditional energy stability for the stabilized semi-implicit time-stepping methods. It is shown that the size of stabilization term depends on the initial energy and the perturba- tion parameter but is independent of the time step. The corresponding error analysis is also established under minimal nonlinearity and regularity assumptions.

Phase Field Modeling of the Zn1-xCd_xO Solid Solutions

The analysis of spinodal decomposition in the Zn1-xCdxO ternary alloy was carried out by means of the nonlinear Cahn-Hilliard equation. Interaction parameter as a function of composition x was provided by valence force field simulations and was used in this analysis. The morphological patterns for the ternary alloys with different Cd content (x = 5, 10, 50%) were experimentally obtained using the semi-implicit Fourier-spectral method. The simulated microstructure evolution Zn0.95Cd0.05O demonstrates that the microstructure having a form of bicontinuous worm-like network is evolved with the progress of aging. An effect of the phase-field mobility and the gradient energy on the microstructure evolution of the Zn1-xCdxO alloys is discussed. It was found that the higher driving force for the decomposition in the higher Cd content film results in a higher decomposition rate revealed by the simulations. The temporal evolution of the simulated Zn0.95Cd0.05O microstructure is in good agreement with experimental results, which have been obtained for this solid solution.

Numerical analysis of energy-minimizing wavelengths of equilibrium states for diblock copolymers

We present a robust and accurate numerical algorithm for calculating energy-minimizing wavelengths of equilibrium states for diblock copolymers. The phase-field model for diblock copolymers is based on the nonlocal Cahn–Hilliard equation. The model consists of local and nonlocal terms associated with short- and long-range interactions, respectively. To solve the phase-field model efficiently and accurately, we use a linearly stabilized splitting-type scheme with a semi-implicit Fourier spectral method. To find energy-minimizing wavelengths of equilibrium states, we take two approaches. One is to obtain an equilibrium state from a long time simulation of the time-dependent partial differential equation with varying periodicity and choosing the energy-minimizing wavelength. The other is to directly solve the ordinary differential equation for the steady state. The results from these two methods are identical, which confirms the accuracy of the proposed algorithm. We also propose a simple and powerful formula: h = L∗/m, where h is the space grid size, L∗ is the energy-minimizing wavelength, and m is the number of the numerical grid steps in one period of a wave. Two- and three-dimensional numerical results are presented validating the usefulness of the formula without trial and error or ad hoc processes.

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.

Simulation of Solid Oxide Fuel Cell Anode Microstructure Evolution Using Phase Field Method

A phase field method is used to simulate the microstructural evolution of nickel-yttria stabilized zirconia composite anode of solid oxide fuel cell based on three-dimensional microstructures reconstructed by focused ion beam-scanning-electron-microscopy technique. By implementing efficient semi-implicit Fourier spectral method in the phase field simulation with different periodic boundary conditions, the computation efficiency is improved in orders of magnitude. In order to quantitatively study the correlation between the anode microstructure evolution and the resulting electrochemical performance degradation, the evolution of three-phase boundary and nickel specific surface area are calculated. Degradation experiments have been conducted to evaluate the simulation results. By choosing an appropriate value for mobility in the simulation, predicted and experimental results showed good agreement.

Simulation of Solid Oxide Fuel Cell Anode Microstructure Evolution Using Phase Field Method

In the present study, a phase field method is used to simulate the microstructural evolution of nickel-yttria stabilized zirconia composite anode of solid oxide fuel cell based on three-dimensional microstructures reconstructed by focused ion beam-scanning-electron-microscopy technique. A three-dimensional reconstruction of the anode sample after initial reduction is used as the initial microstructure. By implementing efficient semi-implicit Fourier spectral method in the phase field simulation with different periodic boundary conditions, the computation efficiency is improved in orders of magnitude. In order to quantitatively study the correlation between the anode microstructure evolution and the resulting electrochemical performance degradation, the evolution of three-phase boundary and nickel specific surface area and the nickel and pore phase tortuosity factors are quantitatively calculated in simulations. For comparison, different real time experiments have been conducted and the simulation results showed good agreement with the real time experimental data.

The Self-Accommodation and Rearrangement of Martensite Multi-Variants under Cyclic Stress: Phase-Field Simulation

A twin boundary model was established to describe the multi-variant interface in the martensitic materials. The modified semi-implicit Fourier-spectral method was proposed to solve the 3-D phase-field equation. Self-accommodation plays an important role in the micro-structural evolution during the loading and unloading. The external compressive stress can cause the rearrangement of martensites from three variants to one variant. After releasing the loading, another variant can nucleate and grow in one variant at the twin boundary. Cyclic stress may lead to the redistribution of martensite variants besides the rearrangement.

Three-dimensional chemotaxis model for a crawling neutrophil

Chemotactic cell migration is a fundamental phenomenon in complex biological processes. A rigorous understanding of the chemotactic mechanism of crawling cells has important implications for various medical and biological applications. In this paper, we propose a three-dimensional model of a single crawling cell to study its chemotaxis. A single-cell study of chemotaxis has an advantage over studies of a population of cells in that it provides a clearer observation of cell migration, which leads to more accurate assessments of chemotaxis. The model incorporates the surface energy of the cell and the interfacial interaction between the cell and substrate. The semi-implicit Fourier spectral method is applied to achieve high efficiency and numerical stability. The simulation results provide the kinetic and morphological traits of a crawling cell during chemotaxis.