Conservative Allen-Cahn Equation Research Articles

Consistent and conservative lattice Boltzmann method for axisymmetric multiphase electrohydrodynamic flows

In this work, we first develop a consistent and conservative mathematical model to study multiphase electrohydrodynamic (EHD) flows, including the reduction-consistent and conservative Allen-Cahn equation for the multiphase field, the Laplace equation for the electric potential, and the consistent and conservative Navier–Stokes equations for the flow field. Owing to the reduction-consistent property, the present model can be used to handle a set of M (1≤M≤N) EHD fluids in N-phase system. Then a two-relaxation-time lattice Boltzmann (LB) method is proposed for axisymmetric multiphase EHD flows, which can correctly recover the axisymmetric governing equations through the direct Taylor expansion analysis. To test the capacity of the LB method, the deformations of a single two-phase droplet and a ternary-phase compound droplet under a uniform electric field are considered. It is found that different from a single droplet with two deformation modes, the compound droplet has four deformation modes, and additionally, the effects of the electric strength, the conductivity ratio and the permittivity ratio on the compound droplet are also investigated. Finally, the compound droplet in the quaternary-phase system is also explored, and there are eight deformation modes that have not been reported in the available literature.

On high-order schemes for the space-fractional conservative Allen–Cahn equations with local and local–nonlocal operators

In this study, we focus on two fractional conservative Allen–Cahn equations with a nonlocal space-independent operator (called the RSLM operator) and a local–nonlocal space–time dependent operator (called the BBLM operator), respectively. Recently, scholars have found that the fractional Allen–Cahn equation is better than the classical equation for describing the interface thickness. Subsequently, the fractional conservative Allen–Cahn equation was used for conservative fluid dynamics. We find that two fractional conservative Allen–Cahn equations both hold the mass conservation, energy stability, and maximum bound principle. In this study, we construct second- and third-order numerical algorithms for two fractional conservative Allen–Cahn equations, and prove the mass conservation and energy decrease laws for them. To our knowledge, for the conservative Allen–Cahn equation, there are only a few works for high-order schemes with proof of their energy stability. We then present several numerical examples whose results show that the convergence orders of the numerical solutions reach the second- and third-order in time as expected. We also simulate some phase transition behaviors to be mass conservative and energy stable, as described by the fractional conservative Allen–Cahn equations, and have some interesting discoveries. It is found that the BBLM formulation can better maintain small features, but the RSLM formulation is better at capturing the profile of the interface. We also compare two fractional conservation equations with the fractional Cahn–Hilliard equation, whose changes are similar to those of the RSLM formulation. In particular, we propose the variable-step BDF2 and BDF3 methods, and prove the mass conservation and energy stability under a time-ratio constraint.

Energetic spectral-element time marching methods for phase-field nonlinear gradient systems

We propose two efficient energetic spectral-element methods in time for marching nonlinear gradient systems with the phase-field Allen–Cahn equation as an example: one fully implicit nonlinear method and one semi-implicit linear method. Different from other spectral methods in time using spectral Petrov-Galerkin or weighted Galerkin approximations, the presented implicit method employs an energetic variational Galerkin form that can maintain the mass conservation and energy dissipation property of the continuous dynamical system. Another advantage of this method is its superconvergence. A high-order extrapolation is adopted for the nonlinear term to get the semi-implicit method. The semi-implicit method does not have superconvergence, but can be improved by a few Picard-like iterations to recover the superconvergence of the implicit method. Numerical experiments verify that the method using Legendre elements of degree three outperforms the 4th-order implicit-explicit backward differentiation formula and the 4th-order exponential time difference Runge-Kutta method, which were known to have best performances in solving phase-field equations. In addition to the standard Allen–Cahn equation, we also apply the method to a conservative Allen–Cahn equation, in which the conservation of discrete total mass is verified. The applications of the proposed methods are not limited to phase-field Allen–Cahn equations. They are suitable for solving general, large-scale nonlinear dynamical systems.

Lattice Boltzmann simulation of breaking waves

Water wave breaking at the ocean’s surface is a turbulent phenomenon and encompasses a wide range of physical processes such as generation of bubbles/drops with different spatio-temporal scales. In the present study, breaking of periodic waves in deep water regime has been simulated using phase-field lattice Boltzmann method (LBM). Compared to direct Navier–Stokes equation based numerical models, the present LB model is Poisson-free and explicit in time integration. Both the flow field and interface are solved using LBM with two sets of distribution functions. The normalized pressure-velocity formulation is used for the flow field, whereas the conservative Allen–Cahn equation is used to capture the interface. An impulsively started wave approach with Stokes third order wave profile has been used to initiate the breaking process. Through systematic variation of Weber number and initial steepness, the effect of surface tension on the kinematic, geometric, and energy dissipation characteristics has been studied for three breaker types, viz., spilling, weak plunging, and plunging breakers. The results are in good agreement with the past experimental observations and direct numerical simulation. In particular, the obtained breaking strength for different Weber numbers agrees well with the semi-empirical relation used in a wind-wave forecasting models.

Maximum bound principle preserving and mass conservative projection method for the conservative Allen–Cahn equation

In this paper, we propose and analyze an efficient maximum bound principle (MBP) preserving and mass conservative projection method for the conservative Allen–Cahn equation. The proposed projection operator can be proven contractive in the discrete L2 norm. Discrete MBP and mass conservation are rigorously presented for a second-order exponential time differencing scheme with a central difference discretization in space. Various numerical examples are performed to verify these theoretical results and demonstrate the accuracy and robustness of the proposed scheme.

A multi-physical structure-preserving method and its analysis for the conservative Allen-Cahn equation with nonlocal constraint

A multi-physical structure-preserving method and its analysis for the conservative Allen-Cahn equation with nonlocal constraint

A deep learning method for the dynamics of classic and conservative Allen-Cahn equations based on fully-discrete operators

The Allen-Cahn equation is a well-known stiff semilinear parabolic equation used to describe the process of phase separation and transition in phase field modeling of multi-component physical systems, while the conservative Allen-Cahn equation is a modified version of the classic Allen-Cahn equation that can additionally conserve the mass. As neural networks and deep learning techniques have achieved significant successes in recent years in scientific and engineering applications, there has been growing interest in developing deep learning algorithms for numerical solutions of partial differential equations. In this paper, we propose a novel deep learning method for predicting the dynamics of the classic and conservative Allen-Cahn equations. Specifically, we design two special convolutional neural network models, one for each of the two equations, to learn the fully-discrete operators between two adjacent time steps. The loss functions of the two models are defined using the residual of the fully-discrete systems, which result from applying the central finite difference discretization in space and the Crank–Nicolson approximation in time. This approach enables us to train the models without requiring any ground-truth data. Moreover, we introduce an effective training strategy that automatically generates useful samples along the time evolution to facilitate training of the models. Finally, we conduct extensive experiments in two and three dimensions to demonstrate outstanding performance of our proposed method, including its dynamics prediction and generalization ability under different scenarios.

A lattice Boltzmann model for the interface tracking of immiscible ternary fluids based on the conservative Allen-Cahn equation

A lattice Boltzmann (LB) model for the interface tracking of immiscible ternary fluids is proposed based on the conservative Allen-Cahn (A-C) equation. A nonlocal equilibrium distribution function source term is adopted in the LB equation to eliminate the unwanted numerical items. With the new LB equation and the carefully designed equilibrium distribution function, Chapman-Enskog analysis shows that the present model can correctly recover the conservative A-C equation with second-order accuracy. A series of benchmark cases have been used to validate the accuracy and stability of the present model, including the classical diagonal translation of two concentric circular interfaces, the rigid-body rotation of Zalesak's disk, and the periodic shear deformation of two tangent circular interfaces. By coupling the dynamic LB model for Navier-Stokes equations, the stationary droplets, the spreading of a liquid lens, the spinodal decomposition, and the Raleigh-Taylor instability have also been simulated in the framework of ternary fluids. Utilizing the geometric interpolation method, the wettability effect of the horizontal boundary is considered and numerical tests with different wall contact angles have been performed to verify the applicability of the present model. All the numerical tests show that the present model can capture the interfaces among the immiscible ternary fluids accurately. Furthermore, using fewer discrete velocity directions, the present model shows higher numerical accuracy and stability compared with other existing numerical models. Therefore, it is promising to improve computational efficiency.

A three-dimensional fully threaded tree adaptive mesh phase-field lattice Boltzmann method for gas–liquid phase change problems

A fully threaded tree adaptive mesh lattice Boltzmann method based on the phase-field model with the conservative Allen–Cahn equation is presented for the simulation of multiphase flows and heat transfer, especially the gas–liquid phase change problems in three dimensions. The presented model incorporates the conservative Allen–Cahn equation for interface tracking and employs hydrodynamics and temperature evolution D3Q19 lattice Boltzmann equations to recover the corresponding Navier–Stokes equations and energy equations. The gas–liquid phase change at the phase interface can be reflected with introducing the mass production rate in the lattice Boltzmann evolution equations. With the fully threaded tree adaptive mesh implemented to capture the phase interface, the computational efficiency can obviously be raised while ensuring the accurate capture of gas–liquid interface. The present method is used to reproduce several classical benchmarks, namely, the droplet evaporation in superheated gas, the buoyancy-driven bubble rising in viscous liquid, the 3-dimensional Rayleigh Taylor instability problem, the nucleate boiling on a wall with constant temperature, and the film boiling on superheated bottom.

A filtering approach for the conservative Allen–Cahn equation solved by the lattice Boltzmann method and a numerical study of the interface thickness

A filtering approach for the conservative Allen–Cahn equation solved by the lattice Boltzmann method and a numerical study of the interface thickness

High-order analysis of lattice Boltzmann models for the conservative Allen-Cahn equation

High-order analysis of lattice Boltzmann models for the conservative Allen-Cahn equation

Gradient-descent-like scheme for the Allen–Cahn equation

The phase-field equations have many attractive characteristics. First, phase separation can be induced by the phase-field equations. It transforms from a single homogeneous mixture to two distinct phases in a nascent state. Second, the solution of the phase-field equations is bounded by a finite value. It is beneficial to ensure numerical stability. Third, the motion of the interface can be described by geometric features. It is helpful for expressing natural phenomena in mathematical terms. Fourth, the phase-field equations possess the energy dissipation law. This law is about degeneration and decay. It tells us in thermodynamics that all occurrences are irreversible processes. In this paper, we would like to investigate the numerical implementation of the Allen–Cahn (AC) equation, which is the classical one of the phase-field equations. In phase field modeling, the binary phase system is described using a continuous variable called the order parameter. The order parameter can be categorized into two forms: conserved, which represents the physical property such as concentration or mass, and non-conserved, which does not have the conserved physical property. We consider both the non-conservative and conservative AC equations. Our interest is more precisely to scrutinize the utilization of the discrete Laplacian operator in the AC equation by considering the conservative and non-conservative order parameter ϕ. Constructing linearly implicit methods for solving the AC equation, we formulate a gradient-descent-like scheme. Therefore, reinterpreting the implicit scheme for the AC equation, we propose a novel numerical scheme in which solutions are bounded by 1 for all t > 0. Together with the conservative Allen–Cahn equation, our proposed scheme is consistent when mass is conserved as well. From a numerical point of view, a linear, unconditionally energy stable splitting scheme is transformed into a gradient-descent-like scheme. Various numerical simulations are illustrated to demonstrate the validity of the proposed scheme. We also make distinctions between the proposed one and existing numerical schemes.

A three-dimensional non-orthogonal multiple-relaxation-time phase-field lattice Boltzmann model for multiphase flows at large density ratios and high Reynolds numbers

This study proposes a three-dimensional non-orthogonal multiple-relaxation-time (NMRT) phase-field multiphase lattice Boltzmann (PFLB) model within a recently established unified lattice Boltzmann model (ULBM) framework [Luo et al., Phil. Trans. R. Soc. A 379, 20200397, 2021]. The conservative Allen-Cahn equation and the incompressible Navier-Stokes (NS) equations are solved. In addition, a local gradient calculation scheme for the order parameter of the Allen-Cahn equation is constructed with the non-equilibrium part of the distribution function. A series of benchmark cases are conducted to validate the proposed model, including the two-phase Poiseuille flow, Rayleigh-Taylor instability, binary liquid/metal droplet collision, and a bubble rise in water. The present simulation results are in good agreement with existing simulation and experimental data. In the simulation of the co-current two-phase Poiseuille flow, the present model is proven to resolve the discontinuity at the phase interface and provide accurate results at extremely high density ratios (i.e., up to 106). Finally, the proposed model is adopted to simulate two challenging cases: (1) water droplet splashing during its impacting on a thin liquid film and (2) liquid jet breakup. The simulation results demonstrate an excellent agreement with previous experimental results, both qualitatively and quantitatively. In these simulations, the Weber number and Reynolds number reach 105 and 6000, respectively, and the viscosity can be as low as ∼10−4, in the lattice unit.

A phase-field Lattice Boltzmann method for liquid-vapor phase change problems based on conservative Allen-Cahn equation and adaptive treegrid

A phase-field Lattice Boltzmann method (LBM) based on Allen-Cahn (A-C) equation with a modified adaptive mesh refinement (AMR) method is presented to simulate the liquid-vapor phase change problem. Three Lattice Boltzmann equations are solved: one for the conservative Allen-Cahn equation, one for the incompressible Navier-Stokes equations, and one for the energy equation. These three equations are written in the form of multiple-relaxation-time, together with the equilibrium and forcing terms using the set of Hermite polynomials. The adaptive mesh refinement technique with quadtree grid data structure is implemented to capture the liquid-vapor interface in simulations, in which the mesh can be locally refined by subdividing blocks identified by user-defined criterion. The Lax-Wendroff finite-difference scheme is adopted to calculate the whole computational domain continuously under nonuniform grids. The present method is validated by the classical Stefan problem and then performed to simulate the film boiling heat transfer. The numerical experiments verify the ability of the proposed method to capture the correct evolution of the interface, guarantee the numerical accuracy and remarkably save the computational cost.

A radial basis function approximation method for conservative Allen–Cahn equations on surfaces

In this paper, we present a meshless radial basis function method to solve conservative Allen–Cahn equation on smooth compact surfaces embedded in R3, which can inherits the mass conservation property. The proposed method is established on the operator splitting scheme. We approximate the surface Laplace–Beltrami operator by an iterative radial basis function approximation method and discretize the diffusion equation in time by the Euler method. The reaction equation containing the nonlinear function is solved analytically. Moreover, to make the mass conservation, we employ a kernel-based quadrature formula to approximate the Lagrange multiplier. The salient feature of the meshless conservative scheme is that it is explicit and more efficient than narrow band methods since few scattered nodes on the surface are adopted in spatial approximation. Several numerical experiments are performed to illustrate the accuracy and the conservation property of the scheme on spheres and other surfaces.

弱圧縮性手法を用いた多孔質体内混相流の大規模計算

The simulation of gas-liquid two-phase flow through a porous media by using multiple GPUs have performed. The ideal parallel efficiency of the weak scaling is achieved by applying weakly compressible method, which can avoid to solve Poisson's equation. We can calculate a wide range of parameters by introducing weakly compressible method. The conservative Allen-Cahn equation is solved to capture the gas-liquid interface by using the finite volume method to conserve the total mass of each phase. The simulations of gas-liquid two-phase flow through two-dimensional porous medium are performed by using our solver by using multi GPUs. Ideal parallelization efficiency is achieved by overlapping computation and communication time.

Maximum principle preserving and unconditionally stable scheme for a conservative Allen–Cahn equation

In this study, we present a novel conservative Allen–Cahn (CAC) equation and its maximum principle preserving and unconditionally stable numerical method. There have been many research works of the numerical methods for the CAC equation. To conserve the total mass, many mathematical models for the CAC equation introduced Lagrange multipliers which are added to the original Allen–Cahn equation. Therefore, some of the methods do not preserve the maximum principle, i.e., it is possible to have values greater than the maximum and smaller than the minimum values of the admissible solutions. In this study, we propose a novel CAC equation with a new Lagrange multiplier which is a power exponent to the concentration so that the maximum principle strictly holds. Furthermore, we describe the proposed numerical algorithm in detail and present several computational experiments to validate the superior performance of the proposed scheme.

A hybrid immersed interface and phase-field-based lattice Boltzmann method for multiphase ferrofluid flow

In this study, we developed a hybrid immersed interface and phase-field lattice Boltzmann (LB) method to simulate multiphase ferrofluid flows. The Laplace equation for the magnetic potential with an interface jump condition was numerically solved using a phase-field-based immersed interface method. The phase interface was tracked by the conservative Allen–Cahn equation. One LB equation was used to capture the interfaces between the two immiscible fluids and another to solve for the hydrodynamic properties. The magnetic stress was treated as an interfacial force at the interface, and thus both the magnetic and capillary forces could be formulated in a unified phase-field method framework. Several typical problems, including a circular cylinder in a uniform magnetic field, deformation of a ferrofluid droplet under a uniform magnetic field, two bubbles merging in the ferrofluid, and a ferrofluid droplet suspension under a shear flow to study the rheological characteristics, were simulated to test the accuracy, applicability, and numerical stability of the present model. The numerical examples demonstrated that the present model is able to capture basic phenomenological features in magnetic multiphase flow problems with high density and high viscosity ratios.

Color-gradient lattice Boltzmann model for immiscible fluids with density contrast.

We present a color-gradient-based lattice Boltzmann model for immiscible fluids with a large density contrast. The model employs the velocity-based equilibrium distribution function, initially proposed for the phase-field-based model by Zu and He [Phys. Rev. E 87, 043301 (2013)1539-375510.1103/PhysRevE.87.043301], with a modification necessary to satisfy the kinematic condition at the interface. Different from the existing color-gradient models, the present model allows to specify interface mobility that is independent of the fluid density ratio. Further, we provide a unified framework, which uses the recursive representation of the lattice Boltzmann equation, to derive the governing equationsof the system. The emergent color dynamics thus obtained, through an analysis of the segregation operator, is shown to obey the locally conservative Allen-Cahn equation. We use a series of benchmarks, which include a stationary drop, a layered Poiseuille flow, translation of a drop under a forced velocity field, the Rayleigh-Taylor instability, and the capillary intrusion test to demonstrate the model's ability in dealing with complex flow problems.

Simplified method for wetting on curved boundaries in conservative phase-field lattice-Boltzmann simulation of two-phase flows with large density ratios

A simplified method is proposed to implement a wetting boundary condition on curved surfaces within the conservative phase-field lattice-Boltzmann (LB) simulation framework. It combines the idea of Huang et al. [“An alternative method to implement contact angle boundary condition and its application in hybrid lattice-Boltzmann finite-difference simulations of two-phase flows with immersed surfaces,” Eur. Phys. J. E 41, 17 (2018)] to find the order parameter on the other side of the wall with the conservative Allen–Cahn equation (CACE) for interface evolution solved by the LB equations. It inherits the advantage of the original method using the Cahn–Hilliard equation to avoid complicated interpolations under different geometries. By using the CACE, the boundary condition for the chemical potential is circumvented (making it more simplified), and the overshooting of the order parameter is also greatly suppressed, enabling it to simulate two-phase flows with solid objects of various shapes and wettabilities at large density and viscosity ratios. Several two-dimensional, axisymmetric, and three-dimensional problems, including some previously studied by experiments, were simulated and the capability of the proposed method is demonstrated through its good agreement with theoretical predictions and/or experimental observations.