Nonlinear Cahn-Hilliard Equation Research Articles

Existence, stability and the number of two-dimensional invariant manifolds for the convective Cahn–Hilliard equation

We study the well-known generalised version of the nonlinear Cahn–Hilliard evolution equation, supplemented with periodic boundary conditions. We study local bifurcations in the vicinity of spatially homogeneous equilibrium states. We show the possibility of the existence of a finite or countable set of equilibrium states of the boundary value problem under study, in the vicinity of which, if appropriate conditions are met, there exist two-dimensional invariant manifolds filled with solutions that are periodic in the evolutionary variable. Moreover, we derive asymptotic formulas for these periodic solutions. Finally, we study the stability of invariant manifolds and the solutions belonging to them.In order to analyse the bifurcation problem, we used methods from the theory of dynamical systems with infinite-dimensional phase, namely the method of invariant manifolds and the method of normal forms.

Diagonalization based Parallel-in-Time method for a class of fourth order time dependent PDEs

In this paper, we design, analyze and implement efficient time parallel methods for a class of fourth order time-dependent partial differential equations (PDEs), namely the biharmonic heat equation, the linearized Cahn–Hilliard (CH) equation and the nonlinear CH equation. We use a diagonalization technique on all-at-once system to develop efficient iterative time parallel methods for investigating the solution behaviour of the said equations. We present the convergence analysis of Parallel-in-Time (PinT) algorithms. We verify our findings by presenting numerical results.

Cahn–Hilliard equation with regularization term

We will study in this article the nonlinear Cahn–Hilliard equation with proliferation and regularization terms with regular and logarithmic potentials. First, we consider the regular potential case, we show that the solutions blow up in finite time or exist globally in time. Furthermore, we prove that the model possess a global attractor. In addition, we construct a robust family of exponential attractors, i.e. attractors which are continuous with respect to the perturbation parameter. In the second part, we consider the logarithmic potential case and show the existence of a global solution.

Mastering the Cahn–Hilliard equation and Camassa–Holm equation with cell-average-based neural network method

In this paper, we develop cell-average-based neural network (CANN) method to approximate solutions of nonlinear Cahn–Hilliard equation and Camassa–Holm equation. The CANN method is motivated by the finite volume scheme and evolved from the integral or weak formulation of partial differential equations. The major idea of cell-average-based neural network method is to explore a neural network to approximate the solution average difference or evolution between two neighboring time steps. Unlike traditional numerical methods, CANN method is not limited by the CFL restriction and can adapt large time steps for solution evolution, which is a significant advantage that classical numerical methods do not have. Once well trained, this method can be implemented as an fixed explicit finite volume scheme and applied to certain groups of initial value conditions for Cahn–Hilliard equation and Camassa–Holm equation without retraining the neural network. Furthermore, the CANN method also performs very well in handling data with corruption or low-quality data generated by Gaussian white noise. Numerical examples are presented to demonstrate effectiveness, accuracy and capability of the proposed method.

The Effect of Conductive Heat Transfer on the Morphology Formation in Polymer Solutions Undergoing Thermally Induced Phase Separation.

Owing to the fact that heat transfer during the thermally induced phase separation process is limited, a quench rate is inevitably entailed, which leads to the existence of temporal and spatial variations in temperature. Hence, it is of great importance to take into account the nonisothermality during the phase separation process, especially in high viscosity polymer solutions. In this study, the influence of conductive heat transfer on the morphology formation during the thermally induced phase separation process was investigated theoretically in terms of quench depth, boundary conditions, and enthalpy of demixing to elucidate the interaction between temperature and concentration through incorporating the nonlinear Cahn-Hilliard equation and the Fourier heat transfer equation in two dimensions. The Flory-Huggins free energy theory for the thermodynamics of phase separation, slow mode theory, and Rouse law for polymer diffusion without entanglements were taken into account in the model development. The simulation results indicated a strong interaction between heat transfer and phase separation, which impacted the morphology formation significantly. Results confirmed that quench depth had an indispensable impact on phase separation in terms of higher characteristic frequency by increasing the driving force for heat transfer. Applying quench from various boundaries led to a difference in the quench rate due to the high viscosity of the polymer solution. This led to a gradation in pore size and anisotropic morphology formation. The degree and direction of anisotropy depended on quench depth and rate, quench time, heat conduction rate inside the solution, solution viscosity, temperature evolution, and the enthalpy of demixing. It was also verified that the influence of enthalpy of demixing on phase separation could not be neglected as it increased the solution temperature and led to phase separation being accomplished at a higher temperature than the initial quench temperature.

Fully decoupled pressure projection scheme for the numerical solution of diffuse interface model of two-phase flow

We describe a fully decoupled finite difference approach for solving the Cahn–Hilliard Navier–Stokes (CHNS) model of two-phase flow. The governing equations consist of the incompressible Navier–Stokes equations coupled with the nonlinear Cahn–Hilliard equation through surface tension. The coupled system of equations is decoupled with the help of an intermediate velocity-field. The spatial discretization is carried out by finite differences while the explicit Euler’s method is used for the temporal part. The decoupling feature of the method makes the numerical implementation easy. Several two-phase flow problems have been numerically simulated in order to test the method’s efficacy.

Towards predictive simulations of spinodal decomposition in Fe-Cr alloys

Simulations of spinodal decomposition in an Fe-36 wt%Cr alloy at 773 K are performed by solving the non-linear Cahn–Hilliard equation, and the results are compared with atom probe tomography measurements. The influence of gradient energy coefficient, atomic mobilities and initial structure on the kinetics of spinodal decomposition is studied. It is shown that a proper initial structure, accounting for the thermal history above the miscibility gap, is crucial and enables predictive simulations of spinodal decomposition in Fe-Cr alloys.

Highly efficient and stable numerical algorithm for a two-component phase-field crystal model for binary alloys

This paper considers the numerical approximation of a two-component phase-field crystal model consisting of two coupled nonlinear Cahn–Hilliard equations of binary alloys. We develop a highly efficient time-marching scheme with second-order accuracy based on the SAV approach, in which two additional stabilization terms are introduced to improve stability, thus allowing large time steps. Unconditional energy stability is then proved strictly. By simulating a large number of numerical simulations in 2D and 3D, including binary crystal growth and phase separation with vacancies, we then verify the stability and accuracy of the scheme.

Phase separation dynamics in aqueous solutions of thermoresponsive polymers

Phase transition kinetics of aqueous hydroxypropyl cellulose solution was studied by using turbidimetric monitoring and mathematical modelling techniques. Based on the nonlinear Cahn-Hilliard equation with a mobility depending on the component concentration, the phase separation has been modeled on a simple one-dimensional Flory lattice. For value set of the interfacial energy parameter, data were obtained on the changing of the average values of the cluster sizes, their mass and concentration. The simulation results allow us to distinguish three stages of the spinodal decomposition: early, intermediate and final. It was found that for the intermediate stage, the kinetics of the cluster mass growth is described by a dependence that is characteristic of the usual diffusion mass transfer; the change in the average cluster size can be represented by a scaling function with an exponent close to 1/3, typical of the systems with a conserved scalar order parameter. It is shown that the concentration of clusters at the final stage is determined by the temperature dependence of the interfacial energy.

Simulation of nonlinear Cahn-Hilliard equation based on local refinement pure meshless method

The phase separation phenomenon between different matters plays an important role in many science fields. And the high order nonlinear Cahn-Hilliard (C-H) equation is often used to describe the phase separation phenomenon between different matters. However, it is difficult to solve the high-order nonlinear C-H equations by the theorical methods and the grid-based methods. Therefore, in this work the meshless methods are addressed, and a local refinement finite pointset method (LR-FPM) is proposed to numerically investigate the high-order nonlinear C-H equations with different boundary conditions. Its constructive process is as follows. 1) The fourth derivative is decomposed into two second derivatives, and then the spatial derivative is discretized by FPM based on the Taylor series expansion and weighted least square method. 2) The local refinement and quintic spline kernel function are employed to improve the numerical accuracy. 3) The Neumann boundary condition with high-order derivatives is accurately imposed when solving the local linear equation sets. The 1D/2D C-H equations with different boundary conditions are first solved to show the ability of the LR-FPM, and the analytical solutions are available for comparison. Meanwhile, we also investigate the numerical error and convergence order of LR-FPM with uniform/non-uniform particle distribution and local refinement. Finally, 1D/2D C-H equation without analytical solution is predicted by using LR-FPM, and compared with the FDM result. The numerical results show that the implement of the boundary value condition is accurate, the LR-FPM indeed has a higher numerical accuracy and convergence order, is more flexible and applicable than the grid-based FDM, and can accurately predict the time evolution of nonlinear diffusive phase separation phenomenon between different materials time.

Code Generation for Generally Mapped Finite Elements

Many classical finite elements such as the Argyris and Bell elements have long been absent from high-level PDE software. Building on recent theoretical work, we describe how to implement very general finite-element transformations in FInAT and hence into the Firedrake finite-element system. Numerical results evaluate the new elements, comparing them to existing methods for classical problems. For a second-order model problem, we find that new elements give smooth solutions at a mild increase in cost over standard Lagrange elements. For fourth-order problems, however, the newly enabled methods significantly outperform interior penalty formulations. We also give some advanced use cases, solving the nonlinear Cahn-Hilliard equation and some biharmonic eigenvalue problems (including Chladni plates) using C 1 discretizations.

Local discontinuous Galerkin methods with implicit-explicit multistep time-marching for solving the nonlinear Cahn-Hilliard equation

In this paper, we develop the fully discrete local discontinuous Galerkin (LDG) methods coupled with the implicit-explicit (IMEX) multistep time marching for solving the nonlinear Cahn-Hilliard equation. To be more specific, we rewrite the Cahn-Hilliard equation in a novel form by adding and subtracting a “linear” term. Then we discretize the spatial derivatives with the LDG methods and the temporal derivative with the IMEX multistep method. Finally, a series of numerical experiments are given to verify the accuracy, efficiency and validness of proposed method.

An operational matrix for solving time-fractional order Cahn-Hilliard equation

In the present scientific work, an operational matrix scheme with Laguerre polynomials is applied to solve a space-time fractional order non-linear Cahn-Hilliard equation, which is used to calculate chemical potential and free energy for a non-homogeneous mixture. Constructing operational matrix for fractional differentiation, the collocation method is applied to convert Cahn-Hilliard equation into an algebraic system of equations, which have been solved using Newton method. The prominent features of the manuscript is to providing the stability analysis of the proposed scheme and the pictorial presentations of numerical solution of the concerned equation for different particular cases and showcasing of the effect of advection and reaction terms on the nature of solute concentration of the considered mathematical model for different particular cases.

A fast immersed interface method for the Cahn–Hilliard equation with arbitrary boundary conditions in complex domains

A fast immersed boundary method for the Cahn–Hilliard equation is introduced. The decomposition of the fourth-order non-linear Cahn–Hilliard equation into a system of linear parabolic second-order equations allows to pose arbitrary Neumann or surface wetting conditions on the boundary. In space a finite-volume discretization on a regular Cartesian voxel grid allows the use of fast parabolic solvers via Fourier transform of arbitrary convergence order. For the time discretization, a second-order Runge–Kutta scheme is applied. The polynomial approximation of the chemical potential results in a numerical scheme that is unconditionally gradient-stable and allows large time steps. With an additional pre-conditioner for the linear system, the condition of linear system is minimized. By this the convergence is independent of both spatial discretization and time step size. This allows for the simulation of phase separation in large porous complex domains with three dimensions and over hundred million degrees of freedom while applying arbitrary boundary conditions and using large time steps.

Analysis of Spinodal Decomposition in Al-Zn and Al-Zn-Cu Alloys Using the Nonlinear Cahn-Hilliard Equation

The phase field model based on the nonlinear Cahn-Hilliard equation was applied to analyze the spinodal decomposition process in Al-Zn and Al-Zn-Cu alloys. Partial differential equations were solved using the explicit finite difference method for the Al- 20, and 35 at. % Zn alloys aged at temperatures between 25 and 100 °C for times from 10 s to 2000 s and Al-20at.%Zn-10at.%Cu and Al-20at.%Zn-5at.%Cu alloys at temperatures between 400 and 500 °C for times from 3600 to 360000 s. Thermo-Calc indicated that the copper addition extends the presence of the metastable miscibility gap up to a temperature of about 597 °C in comparison to the temperature of 350 °C for the binary case. This miscibility gap was calculated assuming that the equilibrium phases were not present and thus it is only existing at the early stages of aging. Simulation results pointed out that the phase decomposition process is much faster in the binary alloys than that in the ternary alloys in spite of the higher aging temperature for the latter case.

Invariant manifolds of interior multi-spike states for the Cahn-Hilliard equation in higher space dimensions

We construct invariant manifolds of interior multi-spike states for the nonlinear Cahn-Hilliard equation and then investigate the dynamics on it. An equation for the motion of the spikes is derived. It turns out that the dynamics of interior spikes has a global character and each spike interacts with all the others and with the boundary. Moreover, we show that the speed of the interior spikes is super slow, which indicates the long time existence of dynamical multi-spike solutions in both positive and negative time. This result is the application of an abstract result concerning the existence of truly invariant manifolds with boundary when one has only approximately invariant manifolds. The abstract result is an extension of one by P. Bates, K. Lu, and C. Zeng to the case of a manifold with boundary, consisting of almost stationary states.

Numerical Simulation of Phase Separation Kinetic of Polymer Solutions Using the Spectral Discrete Cosine Transform Method

ABSTRACTA spectral discrete cosine transform (DCT) method was used to solve numerically the spatio-temporal nonlinear Cahn-Hilliard equation for a temperature-induced phase separation process of a polymer solution. The properties of a high molecular weight polystyrene solution were used in the model to reflect the behavior of a real world polymer system. Based on the value of the initial concentration, three different final morphologies of the phase separated system, granular, interconnected, and microcellular, were identified. It was shown that the initial concentration is the main factor to determine the high rate of the phase separation when the phase separation time is the primary desired parameter. The degree of phase separation, as a quantitative measure of the phase separation process, indicates the rate and amount of separated polymer material for practical applications. The model is capable of providing quantitative information of morphology evolution during phase separation processes for microstructure control purposes. The structure factor profile was extracted as a theoretical output that enables comparison with scattering experiments observations as the finger-print of phase separation morphological evolution. It is shown that the DCT method is a very suitable and feasible method for solution of the nonlinear partial differential equation of phase separation kinetics.

A Decoupled Unconditionally Stable Numerical Scheme for the Cahn–Hilliard–Hele-Shaw System

We propose a novel decoupled unconditionally stable numerical scheme for the simulation of two-phase flow in a Hele-Shaw cell which is governed by the Cahn-Hilliard-Hele-Shaw system (CHHS) with variable viscosity. The temporal discretization of the Cahn-Hilliard equation is based on a convex-splitting of the associated energy functional. Moreover, the capillary forcing term in the Darcy equation is separated from the pressure gradient at the time discrete level by using an operator-splitting strategy. Thus the computation of the nonlinear Cahn-Hilliard equation is completely decoupled from the update of pressure. Finally, a pressure-stabilization technique is used in the update of pressure so that at each time step one only needs to solve a Poisson equation with constant coefficient. We show that the scheme is unconditionally stable. Numerical results are presented to demonstrate the accuracy and efficiency of our scheme.

A Study of Some Nonlinear Partial Differential Equations by Using Adomian Decomposition Method and Variational Iteration Method

In this paper, a numerical solution of nonlinear partial differential equation, Benjamin-Bona-Mahony (BBM) and Cahn-Hilliard equation is presented by using Adomain Decomposition Method (ADM) and Variational Iteration Method (VIM). The results reveal that the two methods are very effective, simple and very close to the exact solution.

Approximate Solutions and Symmetry Group for Initial-Value Problem of Nonlinear Cahn-Hilliard Equation

Approximate Solutions and Symmetry Group for Initial-Value Problem of Nonlinear Cahn-Hilliard Equation