Modified Cahn Hilliard Equation Research Articles

An improved multiphase lattice Boltzmann flux solver with a modified Cahn–Hilliard equation for multiphase flow with super large density ratio

In this study, a modified Cahn–Hilliard equation with a very simple format was proposed, which can be used to simulate immiscible multi-component/multiphase flow with a super large density ratio. In addition, based on this modified equation and the Navier–Stokes equations, an improved multiphase lattice Boltzmann flux solver (IMLBFS) has been proposed, and its computational ability has been tested by multiple numerical examples, including Laplace law, two bubbles merging, contact angle, bubble rising, and droplet splashing on a thin film. The results show that the proposed IMLBFS can simulate immiscible two-phase flow with a very large density ratio up to 1:5000 or 1:10 000 under various operating conditions, including the Reynolds number reaching 10 000. In addition, IMLBFS also has excellent features such as clear physical properties, freely adjustable source term strength, and effective suppression of mass loss.

A Decoupled Energy Stable Numerical Scheme for the Modified Cahn–Hilliard–Hele–Shaw System with Logarithmic Potential

A decoupled unconditionally stable numerical scheme for the modified Cahn–Hilliard–Hele–Shaw system with logarithmic potential is proposed in this paper. Based on the convex-splitting of the associated energy functional, the temporal discretization of the scheme is given. The fractional step method is used to decouple the nonlinear modified Cahn–Hilliard equation from the pressure gradient. Then, at each time step, one only needs to solve Poisson’s equation which is obtained by using an incremental pressure-stabilization technique. In terms of logarithmic potential, using the regularization procedure can make the domain extended from − 1,1 to − ∞ , ∞ for F ϕ . Finally, the proof of the energy stability and the calculation of the error estimate are presented. Numerical examples are recorded to illustrate the accuracy and performance for the proposed scheme.

Modeling and simulation of multi-component immiscible flows based on a modified Cahn–Hilliard equation

In this study, an efficient method will be developed for the phase-field model of multi-component immiscible phases. The formulation of surface tension requires the interfaces to satisfy the hyperbolic tangent property. However, the interfacial transitions between different phases are not hyperbolic tangent profiles. The enclosed area is not preserved although the total mass is conserved by the original Cahn–Hilliard equation. This study is an extended research based on our previous study (Li et al., 2016). This work aims to apply the modified Cahn–Hilliard model into the multi-phase system. The interface is forced to be hyperbolic tangent by the modified Cahn–Hilliard system. The computational accuracy of the surface tension is improved under our multiphase framework. The mass loss of each phase can be reduced and the enclosed area can be preserved by the proposed method. We show various numerical results to demonstrate the robustness of the proposed modified model.

A second order linear energy stable numerical method for the Cahn–Hilliard–Hele–Shaw system

In this paper, we consider the numerical approximations for the Cahn–Hilliard–Hele–Shaw system, which is a modified Cahn–Hilliard equation coupled with the Darcy flow law. One of challenges in solving this system numerically is how to develop linear discretization for the nonlinear term, while preserving the energy stability. By introducing a Lagrange multiplier r, we construct a fully discrete, second-order linear scheme. The proposed scheme is rigorously proven to be uniquely solvable and unconditionally stable in energy. Moreover, we show theoretical analysis on error estimates of the time step size τ and space step size h at the discrete level. Several numerical examples are presented to demonstrate the stability, accuracy and efficiency of the proposed scheme. We also show numerical simulations on the coarsening dynamics.

Enhancement of damaged-image prediction through Cahn-Hilliard image inpainting.

We assess the benefit of including an image inpainting filter before passing damaged images into a classification neural network. We employ an appropriately modified Cahn–Hilliard equation as an image inpainting filter which is solved numerically with a finite-volume scheme exhibiting reduced computational cost and the properties of energy stability and boundedness. The benchmark dataset employed is Modified National Institute of Standards and Technology (MNIST) dataset, which consists of binary images of handwritten digits and is a standard dataset to validate image-processing methodologies. We train a neural network based on dense layers with MNIST, and subsequently we contaminate the test set with damages of different types and intensities. We then compare the prediction accuracy of the neural network with and without applying the Cahn–Hilliard filter to the damaged images test. Our results quantify the significant improvement of damaged-image prediction by applying the Cahn–Hilliard filter, which for specific damages can increase up to 50% and is advantageous for low to moderate damage.

A weighted essentially nonoscillatory‐based phase field lattice Boltzmann method for incompressible two‐phase flows with high density contrast

AbstractIn this article, a high order weighted essentially nonoscillatory finite difference‐based phase field lattice Boltzmann method (WENO‐PFLBM) is proposed for simulations of incompressible two‐phase flows with high density contrast. The weighted essentially nonoscillatory finite difference scheme is applied to discretize the convection term of the discrete Boltzmann equation for flow field. Moreover, the WENO scheme is also adopted to discretize the convection term of the modified Cahn–Hilliard equation for interface tracking. Numerical validations of the proposed WENO‐PFLBM are implemented by simulating stationary droplet, layered Poiseuille flow, Rayleigh–Taylor instability, bubble rising, and droplet impact on a thin film. Various numerical challenges like high density ratios (up to 1000), complex interfaces, and high Reynolds numbers are included in these examples, which demonstrate the robustness of the present method. In addition, the current method can achieve relatively small spurious velocity compared with the LB‐based model owing to the high order accuracy.

Fast algorithm based on TT-M FE method for modified Cahn–Hilliard equation

We consider numerical methods for solving the modified Cahn–Hilliard equation involving strong nonlinearities. A fast algorithm based on time two-mesh (TT-M) finite element (FE) scheme is proposed to overcome the time-consuming computation of the nonlinear terms. The TT-M FE algorithm includes three main steps: Firstly, a nonlinear FE scheme is solved on a coarse time-mesh τc. Here, the FE method is used for spatial discretization and the implicit second-order 𝜃 scheme (containing both implicit Crank–Nicolson scheme and second-order backward difference method) is used for temporal discretization. Secondly, the Lagrange’s interpolation is used to obtain the interpolation result on the fine time-mesh. Finally, a linearized FE system is solved on a fine time-mesh τ (τ<τc). The stability analysis and priori error estimates are provided in detail. Numerical examples are given to demonstrate the validity of the proposed scheme. The TT-M FE method is compared with the traditional Galerkin FE method and it is evident that the TT-M FE method can save the calculation time.

Modeling and simulation of droplet evaporation using a modified Cahn–Hilliard equation

In this paper, we propose a mathematical model, its numerical scheme, and some computational experiments for droplet evaporation. In order to model the evaporation, a classical Cahn–Hilliard equation with an interfacial evaporation mass flux term is proposed. An unconditionally gradient stable scheme is used to discretize the governing equation, and the multigrid method is applied to solve the resulting system. The proposed model is first validated via a proper interfacial parameter ϵ, and then, the effect of evaporation rate and effect of contact angle on volume and surface area changes are investigated. The numerical results indicate that the dynamics of evaporation are dependent on the contact angle on a solid substrate.

Analysis of a novel finite element method for a modified Cahn–Hilliard–Hele–Shaw system

In this paper, a novel finite element method for solving a modified Cahn–Hilliard–Hele–Shaw system is proposed. The time discretization is based on the convex splitting of the energy functional in the modified Cahn–Hilliard equation, i.e., the high-order nonlinear term and the linear term in the chemical potential are treated explicitly and implicitly, respectively. Designing in this way leads to solving a linear system at each time step, which is much efficient compared to solving a nonlinear system resulting from most existing schemes. The proposed scheme is proved to be unconditionally energy stable and optimally convergent for the phase variable. Numerical results are presented to support our theoretical analysis.

A Large Time-Stepping Mixed Finite Method of the Modified Cahn–Hilliard Equation

The main purpose of this paper is to solve the modified Cahn–Hilliard equation via a large time-stepping mixed finite-element method to ease the time-stepping limit caused by small parameters and nonlinear terms. The modified Cahn–Hilliard equation is discretized by mixed finite-element method in space and first-order semi-implicit scheme in time. The energy stability and error analysis of the fully discrete semi-implicit scheme are proved. Finally, a series of numerical experiments are presented to verify the conclusion of theoretical part.

On the total mass conservation and the volume preservation in the diffuse interface method

In this paper a diffuse interface method which can conserve the total mass and preserve the volume of each phase is proposed for simulating the incompressible two-phase flows. The basic idea of this method is that a source term with two free parameters is introduced into the Cahn–Hilliard equation. The two free parameters are determined by the mass conservation law and the volume preservation property. In the numerical implementation, the multiple-relaxation-time lattice Boltzmann scheme is used to solve the modified Cahn–Hilliard equation and the velocity-based Navier–Stokes equations. To verify the proposed method, several numerical examples are simulated. The numerical results show that the present method can preserve the total mass and the volume of each phase at the same time. The numerical results are also in good agreement with the analytical solutions and the previous results.

Investigation on interlayer phase morphology evolution of “Ex‐Situ” toughened composite laminate

The numerical simulation on the interlayer phase morphology evolution of “Ex‐Situ” toughened laminate was performed in this study. Specifically, the diffusion process of thermoset (TS) resin into thermoplastic (TP) resin was modeled incorporating the negative effect of resin curing reactions. The dynamic process of polymerization‐induced phase separation in TP/TS blends was predicted by the modified Cahn–Hilliard equation, in which the concentration‐dependent mobility was used to describe dynamic asymmetry between TP resin and TS resin, and the mixing free energy of the blends contained the contribution of resin curing reactions. The phase morphology evolution of polysulfone/epoxy blends in the interlaminar region was analyzed numerically, and the results indicated that the diffusion process resulted in the concentration‐gradient distribution of epoxy resin in interlayer, and when the concentration‐gradient decreased, the interlayer phase morphology developed from the coexistence of different microstructures to the presence of a single structural form. The post‐curing temperature played a more important role in the phase separation than that in resin curing reaction, and the separated structures were only observed in the local region of interlayer under low post‐curing temperature, due to small driving force of phase separation. Moreover, the movements of epoxy molecules dominated the microstructure formation process. POLYM. COMPOS., 40:3644–3656, 2019. © 2019 Society of Plastics Engineers

Numerical simulation of pressure-driven phase-change in two-phase fluid flows using the Lattice Boltzmann Method

The main purpose of the current article is to model the phase-change phenomena within a two-phase liquid-gas flow caused by the pressure variations throughout the computational domain. In industrial applications, this phenomenon is mostly known as the cavitation. A lattice Boltzmann framework with two distribution functions is developed in which one of the distribution functions is used to compute the velocity and hydrodynamic pressure while the other one recovers the modified Cahn–Hilliard equation. Proper forcing terms are applied to the governing equations to physically account for the phase transition due to the pressure variation. Moreover, the density of the gaseous phase may change according to the pressure rise/drop. Firstly, a simple benchmark problem which consists of a cylinder containing a mixture of liquid-vapor at the saturation condition is considered. The mixture quality can be varied due to the pressure rise/drop. The thermodynamic and finite difference solution of the problem is successfully compared with the numerical results. Finally the model is extended to two dimensions to simulate the droplet evaporation and condensation as well as the bubble growth and shrinkage.

A PDE Based Image Segmentation Using Fourier Spectral Method

A PDE based binary image segmentation model using a modified Cahn–Hilliard equation with weaker fidelity parameter ( $$\lambda $$ ) and double well potential has been introduced. The threshold of separation $$\gamma $$ is flexibly chosen between 0 and 1. Convexity splitting is used for time discretization and then Fourier-spectral method is used to solve the proposed modified Cahn–Hilliard equation. The proposed model is tested on bio-medical images.

Curve and Surface Smoothing Using a Modified Cahn-Hilliard Equation

We present a new method using the modified Cahn-Hilliard (CH) equation for smoothing piecewise linear shapes of two- and three-dimensional objects. The CH equation has good smoothing dynamics and it is coupled with a fidelity term which keeps the original given data; that is, it does not produce significant shrinkage. The modified CH equation is discretized using a linearly stable splitting scheme in time and the resulting scheme is solved by using a Fourier spectral method. We present computational results for both curve and surface smoothing problems. The computational results demonstrate that the proposed algorithm is fast and efficient.

Pattern formation in the wake of triggered pushed fronts

Pattern-forming fronts are often controlled by an external stimulus which progresses through a stable medium at a fixed speed, rendering it unstable in its wake. By controlling the speed of excitation, such stimuli, or ‘triggers’, can mediate pattern forming fronts which freely invade an unstable equilibrium and control which pattern is selected. In this work, we analytically and numerically study when the trigger perturbs an oscillatory pushed free front. In such a situation, the resulting patterned front, which we call a pushed trigger front, exhibits a variety of phenomenon, including snaking, non-monotonic wave-number selection, and hysteresis. Assuming the existence of a generic oscillatory pushed free front, we use heteroclinic bifurcation techniques to prove the existence of trigger fronts in an abstract setting motivated by the spatial dynamics approach. We then derive a leading order expansion for the selected wave-number in terms of the trigger speed. Furthermore, we show that such a bifurcation curve is governed by the difference of certain strong-stable and weakly-stable spatial eigenvalues associated with the decay of the free pushed front. We also study prototypical examples of these phenomena in the cubic-quintic complex Ginzburg Landau equation and a modified Cahn–Hilliard equation.

A phase-field fluid modeling and computation with interfacial profile correction term

We present a new phase-field fluid model and computation with minimized Cahn–Hilliard (CH) dynamics. Using the CH equation, the internal structure of the interface layer is determined by explicit smoothing flow discontinuities. This method greatly simplifies gridding, discretization, and handling of topological changes. The original CH equation, however, has intrinsic dynamics such as interface length minimization, i.e., the motion by minus the Laplacian of the mean curvature. When the CH equation is applied to the modeling of multiphase fluid flows, we want to minimize its interface length minimization property. The surface tension formulation also requires the multiphase fluid interface to be a hyperbolic tangent profile. Typically, under the advection of flow, the interfacial transition is not a hyperbolic tangent profile, i.e., it is too compressed or sharpened. Even though the original CH dynamics conserves the total mass, the enclosed area obtained by its interface is not preserved. To overcome these shortcomings, we propose a modified CH equation with an interfacial profile correction term. Several numerical examples are presented to show the accuracy of the proposed method. The numerical results demonstrate that the proposed modified CH equation preserves the enclosed area better than the original CH equation.

An efficient numerical method for evolving microstructures with strong elastic inhomogeneity

In this paper, we consider a fast and efficient numerical method for the modified Cahn–Hilliard equation with a logarithmic free energy for microstructure evolution. Even though it is physically more appropriate to use a logarithmic free energy, a quartic polynomial approximation is typically used for the logarithmic function due to a logarithmic singularity. In order to overcome the singularity problem, we regularize the logarithmic function and then apply an unconditionally stable scheme to the Cahn–Hilliard part in the model. We present computational results highlighting the different dynamic aspects from two different bulk free energy forms. We also demonstrate the robustness of the regularization of the logarithmic free energy, which implies the time-step restriction is based on accuracy and not stability.

A mass-conserved diffuse interface method and its application for incompressible multiphase flows with large density ratio

In this work a mass-conserved diffuse interface method is proposed for simulating incompressible flows of binary fluids with large density ratio. In the method, a mass correction term is introduced into the Cahn–Hilliard equation to compensate the mass losses or offset the mass increases caused by the numerical and modeling diffusion. Since the mass losses or increases are through the phase interfaces and at each time step, their values are very small, to keep mass conservation, mass sources or sinks are introduced and uniformly distributed in the volume of diffuse layer. With the uniform distribution, the mass correction term representing mass sources or sinks is derived analytically by applying mass conservation principle. By including the mass correction, the modified Cahn–Hilliard equation is solved by the fifth-order upwind scheme to capture the phase field of the bindery fluids. The flow field is simulated by the newly-developed multiphase lattice Boltzmann flux solver [20]. The proposed approach is validated by simulating the Laplace law, the merging of two bubbles, Rayleigh–Taylor instability and bubble rising under gravity with density ratio of 1000 and viscosity ratio of 100. Numerical results of interface shapes and flow properties agree well with both analytical solutions and benchmark data in the literature. Numerical results also show that the mass is well-conserved in all cases considered. In addition, it is demonstrated that the mass correction term at each time step is in the order of 10−4∼10−5, which is a small number compared with the magnitude of order parameter.

Three-dimensional volume reconstruction from slice data using phase-field models

We propose the application of a phase-field framework for three-dimensional volume reconstruction using slice data. The proposed method is based on the Allen–Cahn and Cahn–Hilliard equations, and the algorithm consists of two steps. First, we perform image segmentation on the given raw data using a modified Allen–Cahn equation. Second, we reconstruct a three-dimensional volume using a modified Cahn–Hilliard equation. In the modified Cahn–Hilliard equation, a fidelity term is introduced to keep the solution close to the slice data. The numerical methods use a hybrid method and an unconditionally stable nonlinear splitting scheme. The resulting discrete equations are solved using a multigrid method. The experiments on synthetic and real medical images are performed to demonstrate the accuracy and efficiency of the proposed method.