Allen-Cahn Type Research Articles

Original energy dissipation preserving corrections of integrating factor Runge-Kutta methods for gradient flow problems

Explicit integrating factor Runge-Kutta methods are attractive and popular in developing high-order maximum bound principle preserving time-stepping schemes for Allen-Cahn type gradient flows. However, they always suffer from the non-preservation of steady-state solution and original energy dissipation law. To overcome these disadvantages, some new integrating factor methods are developed by using two classes of difference correction, including the telescopic correction and nonlinear-term translation correction, enforcing the preservation of steady-state solution. Then the original energy dissipation properties of the new methods are examined by using the associated differential forms and the differentiation matrices. As applications, some new integrating factor Runge-Kutta methods up to third-order maintaining the original energy dissipation law are constructed by applying the difference correction strategies to some popular explicit integrating factor methods in the literature. Extensive numerical experiments are presented to support our theory and to demonstrate the improved performance of new methods.

Asymptotics for vectorial Allen–Cahn type problems

Asymptotics for vectorial Allen–Cahn type problems

A conservative Allen–Cahn model for a hydrodynamics coupled phase-field surfactant system

In this study, we develop a model for a binary fluid–surfactant system utilizing a coupling of two kinds conservative Allen–Cahn type equations and the Navier–Stokes equations. To ensure mass conservation, we incorporate hybrid Lagrange multipliers into the two Allen–Cahn type equations. Specifically, for the concentration variable, a global correction using a time-dependent Lagrange multiplier is utilized, while for the binary fluid variable, a space–time dependent Lagrange multiplier is applied to minimize the impact of dynamics of motion by mean curvature. We propose a linear second order scheme for practical solution of the model. Computational tests demonstrate that the proposed model is effective for the binary fluid–surfactant system and is capable of preserving the small features of interfaces.

A second-order Strang splitting scheme for the generalized Allen–Cahn type phase-field crystal model with FCC ordering structure

In this paper, we consider the generalized Allen–Cahn-type phase-field crystal model with face-centered-cubic ordering structure (PFC-FCC). Due to the combined complexity of the eighth-order spatial derivative and inherent nonlinearity, it poses a significant challenge to design a numerical scheme of high accuracy, stability, and efficiency to solve the PFC-FCC model. Endeavoring towards this objective, we adopt operator splitting method to address the PFC-FCC model. Based on Fourier spectral method for spatial discretization and SSP-RK method for temporal discretization, we propose an effective and easy-to-implement second-order scheme. We are also able to proffer an analytical proof for discrete mass conservation, and engage a discussion on an optimal error estimate for the second-order scheme. In addition, the energy stability of the numerical scheme is derived. Ultimately, a series of numerical experiments specifically designed to substantiate its accuracy, efficiency, and capability for phase transition are performed.

Phase field modeling of hyperelastic material interfaces –Theory, implementation and application to phase transformations

Interface mechanics can significantly govern the evolution of multiple phases on smaller scales, e.g., determining the properties of TWIP- and TRIP-steels, geopolymers or Li-ion batteries. The present contribution is specifically centered around the influence of interface elasticity on mechanically induced phase transformations. A geometrically exact finite element framework is developed for this purpose based on incremental energy minimization. It is set up in three-dimensional space and comprises an Allen-Cahn type phase field formulation based on a Modica-Mortola double-well functional, as well as a Ginzburg-Landau type viscosity formulation. The phase field approximation of the interface deformation gradient is derived by means of homogenization theory. A monolithic solver is employed for the resulting set of non-linear coupled equations. The example of coalescing spheres shows that larger interface energies can amplify and accelerate coalescence. Lower interface energies, in contrast, allow for distinct spreading of the energetically favorable bulk phase. Deformation-dependent interface energies particularly add stress peaks to this process that are localized at the phase transition zone. The relaxation of a stepped interface shape showed that deformation-dependent interface energies favor a plane partition of the bulk phases. Yet, they also delay breakup events compared to simple constant interface energies. This peculiarity can be beneficial for calibration purposes. Strain dependence moreover causes an additional elastic stiffness in the direction of the interface tangent plane and constitutes thus another origin for anisotropy of the overall structure.

The Sharp Interface Limit of an Ising Game

The Ising model of statistical physics has served as a keystone example of phase transitions, thermodynamic limits, scaling laws, and many other phenomena and mathematical methods. We introduce and explore an Ising game, a variant of the Ising model that features competing agents influencing the behavior of the spins. With long-range interactions, we consider a mean-field limit resulting in a nonlocal potential game at the mesoscopic scale. This game exhibits a phase transition and multiple constant Nash-equilibria in the supercritical regime. Our analysis focuses on a sharp interface limit for which potential minimizing solutions to the Ising game concentrate on two of the constant Nash-equilibria. We show that the mesoscopic problem can be recast as a mixed local/nonlocal space-time Allen-Cahn type minimization problem. We prove, using a Γ-convergence argument, that the limiting interface minimizes a space-time anisotropic perimeter type energy functional. This macroscopic scale problem could also be viewed as a problem of optimal control of interface motion. Sharp interface limits of Allen-Cahn type functionals have been well studied. We build on that literature with new techniques to handle a mixture of local derivative terms and nonlocal interactions. The boundary conditions imposed by the game theoretic considerations also appear as novel terms and require special treatment.

Reversible evolution phenomenon of particle during crystal growth: A phase-field study

The two-dimensional thermodynamically consistent phase field model, which consists of the anisotropic Allen-Cahn type equation and the heat equation, is used to study the morphological pattern of the particle in the initial stage of crystal growth. The results reveal the locally reversible growth of the particle in the initial stage of crystal growth. Specifically, some parts of the particle interface first grow inward, while others grow outward from the interface of critical nucleation, until the growth speed of the interface becomes zero. After this, the inward growth parts start to grow outward with other parts. The locally reversible growth of the particle leads to the development of a petal-like shape. The particle radius size dependence of the temperatures along the parts of the particle interface in different crystallographic directions at different time points is studied. By a series of results, the correlations between the morphological pattern selection of the particle and the model parameters, including undercooling and anisotropic strength, are analyzed. These results contribute to understanding the fundamental mechanism of the particle evolution during crystal growth.

Efficient numerical simulation of the conserved Allen–Cahn type flow-coupled binary fluid-surfactant model by a dimension splitting method

Efficient numerical simulation of the conserved Allen–Cahn type flow-coupled binary fluid-surfactant model by a dimension splitting method

A Cahn–Hilliard phase field model coupled to an Allen–Cahn model of viscoelasticity at large strains

We propose a new Cahn–Hilliard phase field model coupled to incompressible viscoelasticity at large strains, obtained from a diffuse interface mixture model and formulated in the Eulerian configuration. A new kind of diffusive regularization, of Allen–Cahn type, is introduced in the transport equation for the deformation gradient, together with a regularizing interface term depending on the gradient of the deformation gradient in the free energy density of the system. The designed regularization preserves the dissipative structure of the equations. We obtain the global existence of a weak solution in three space dimensions and for generic nonlinear elastic energy densities with polynomial growth, comprising the relevant cases of polyconvex Mooney–Rivlin and Ogden elastic energies. Also, our analysis considers elastic free energy densities which depend on the phase field variable and which can possibly degenerate for some values of the phase field variable. We also propose two kinds of unconditionally energy stable finite element approximations of the model, based on convex splitting ideas and on the use of a scalar auxiliary variable respectively, proving the existence and stability of discrete solutions. We finally report numerical results for different test cases with shape memory alloy type free energy, showing the interplay between phase separation and finite elasticity in determining the topology of stationary states with pure phases characterized by different elastic properties.

An efficient linear and unconditionally stable numerical scheme for the phase field sintering model

In this article, the phase field sintering model, which is composed of a Cahn–Hilliard type equation and several Allen–Cahn type equations, has been considered. On the scalar auxiliary variable framework, we propose a theoretically efficient and stable method for solid-state sintering. In order to overcome the nonlinear issues, we define a stabilized scalar auxiliary variable method and reformulate the phase field sintering model. An efficient and accurate numerical scheme is investigated to solve our model. The scheme consist of several decoupled diffusion equations at every time step. Therefore, our scheme is easy to implement. Then we prove the numerical discrete energy is unconditionally stable. Several numerical simulations in two- and three-dimensional spaces are presented to demonstrate the robustness of our method.

Thermodynamically consistent hydrodynamic phase-field computational modeling for fluid-structure interaction with moving contact lines

This paper proposes a novel computational modeling approach to investigate the fluid-structure interactions with moving contact lines. By embracing the generalized Onsager principle, a coupled hydrodynamics and phase field system is introduced that can describe the fluid-structure interactions with moving contact lines in a thermodynamically consistent manner. The resulting partial differential equation (PDE) model consists of the Navier-Stokes equation and a nonlinear Allen-Cahn type equation. Volume conservation is enforced through an additional penalty term. A fully-discrete structure-preserving numerical scheme is proposed by combining several techniques to solve this coupled PDE system effectively and accurately. For the temporal discretization, we utilize the supplementary variable method for preserving the thermodynamic structure and the projection approach for reducing the problem size. Then, we use the finite difference method on the staggered grid for spatial discretization. Furthermore, we have rigorously proved that the proposed numerical scheme based on the second-order backward difference formula respects the original energy stability, i.e., the scheme is energy stable. Additionally, with the aid of the supplementary variable method, the resultant scheme can be transformed into a constrained optimization problem, where the solutions of the supplementary variables are the arguments of the objective function that reaches the optimality. Then the augmented Lagrangian method is introduced in part to bring robustness and efficiency to solving such a constrained optimization problem. Finally, various numerical simulations verify the model's capability and demonstrate the scheme's effectiveness, accuracy, and stability.

On the reaction–diffusion type modelling of the self-propelled object motion

In this study, we propose a mathematical model of self-propelled objects based on the Allen–Cahn type phase-field equation. We combine it with the equation for the concentration of surfactant used in previous studies to construct a model that can handle self-propelled object motion with shape change. A distinctive feature of our mathematical model is that it can represent both deformable self-propelled objects, such as droplets, and solid objects, such as camphor disks, by controlling a single parameter. Furthermore, we demonstrate that, by taking the singular limit, this phase-field based model can be reduced to a free boundary model, which is equivalent to the L^2-gradient flow model of self-propelled objects derived by the variational principle from the interfacial energy, which gives a physical interpretation to the phase-field model.

Highly conservative Allen–Cahn-type multi-phase-field model and evaluation of its accuracy

Abstract In the engineering field, it is necessary to construct a numerical model that can reproduce multiphase flows containing three or more phases with high accuracy. In our previous study, by extending the conservative Allen–Cahn (CAC) model, which is computationally considerably more efficient than the conventional Cahn–Hilliard (CH) model, to the multiphase flow problem with three or more phases, we developed the conservative Allen–Cahn type multi-phase-field (CAC–MPF) model. In this study, we newly construct the improved CAC–MPF model by modifying the Lagrange multiplier term of the previous CAC–MPF model to a conservative form. The accuracy of the improved CAC–MPF model is evaluated through a comparison of five models: three CAC–MPF models and two CH–MPF models. The results indicate that the improved CAC–MPF model can accurately and efficiently perform simulations of multiphase flows with three or more phases while maintaining the same level of volume conservation as the CH model. We expect that the improved CAC–MPF model will be applied to various engineering problems with multiphase flows with high accuracy. Graphic abstract

A modified and efficient phase field model for the biological transport network

This paper aims to establish the biological transport network based on the phase field model. In order to ensure that the topological shape is formed under the guidance of electrical conductivity, we generate the biological networks with sufficient information based on the distributions of the venation of the leaf represented by the reaction-diffusion model. We modify the original energy of the network generating model by considering the auxin gradient property. By applying the gradient flow method to minimize the modified energy, we derive the Poisson type equation for pressure, the reaction-diffusion type equation for the network conductance, and the Allen-Cahn type equation for the phase field. The proposed model is significant on the investigation of phase transitions by considering the gradient properties on the boundaries. We have innovatively added conductivity and phase-field coupling terms that inhibit perpendicular transport of nutrients, making it easy to generate thin branches from the trunk through this model. In order to obtain the second-order temporal accuracy, we take the Crank–Nicolson method for the governing system. To obtain the second-order spatial accuracy, we discretize the coupling system with the central finite difference method and linearize the nonlinear terms semi-explicitly to form a linear system at each time step. The discrete energy dissipation is provably preserved and we can use a larger time step. We apply the preconditioned conjugate gradient method with the multigrid method as a preconditioner to implement a practical algorithm with only linear algebraic complexity. The proposed algorithm is easy to implement and achieves a fast convergence. Various numerical tests are demonstrated to verify the efficiency, stability, and robustness of the proposed method.

Weak convergence of the backward Euler method for stochastic Cahn–Hilliard equation with additive noise

We prove a weak rate of convergence of a fully discrete scheme for stochastic Cahn–Hilliard equation with additive noise, where the spectral Galerkin method is used in space and the backward Euler method is used in time. Compared with the Allen–Cahn type stochastic partial differential equation, the error analysis here is much more sophisticated due to the presence of the unbounded operator in front of the nonlinear term. To address such issues, a novel and direct approach has been exploited which does not rely on a Kolmogorov equation but on the integration by parts formula from Malliavin calculus. To the best of our knowledge, the rates of weak convergence are revealed in the stochastic Cahn–Hilliard equation setting for the first time.

A new Lagrange multiplier method for the mass-conserved Allen–Cahn type square phase-field crystal model

For gradient flows, it is always expected that the two important properties of mass conservation and energy dissipation can be maintained simultaneously. In this paper, we propose a mass-conserved Allen–Cahn type square phase-field crystal (AC-SPFC) model by adding a nonlocal Lagrange multiplier. On the other hand, we construct some linear schemes that dissipate the original energy based on the recently introduced new Lagrange multiplier method (Cheng, Liu and Shen 2020). Noticeably, the new approach does not require a lower bound on the nonlinear part of the energy as the scalar auxiliary variable (Shen, Xu and Yang 2018) method does, and the solving process is pretty straightforward. We exhibit the robustness of the mass-conserved AC-SPFC by comparison with the Cahn–Hilliard type square phase-field crystal equation, and numerically demonstrate the stability and the accuracy of the proposed schemes.

Programming shape-morphing electroactive polymers through multi-material topology optimisation

This paper presents a novel engineering strategy for the design of Dielectric Elastomer (DE) based actuators, capable of attaining complex electrically induced shape morphing configurations. In this approach, a multilayered DE prototype, interleaved with compliant electrodes spreading across the entire faces of the DE, is considered. Careful combination of several DE materials, characterised by different material properties within each of the multiple layers of the device, is pursued. The resulting layout permits the generation of a heterogenous electric field within the device due to the spatial variation of the material properties within the layers and across them. An in-silico or computational approach has been developed in order to facilitate the design of new prototypes capable of displaying predefined electrically induced target configurations. Key features of this framework are: (i) use of a standard two-field Finite Element implementation of the underlying partial differential equations in reversible nonlinear electromechanics, where the unknown fields ot the resulting discrete problem are displacements and the scalar electric potential; (ii) introduction of a novel phase-field driven multi-material topology optimisation framework allowing for the consideration of several DE materials with different material properties, favouring the development of heterogeneous electric fields within the prototype. This novel multi-material framework permits, for the first time, the consideration of an arbitrary number of different N DE materials, by means of the introduction of N−1 phase-field functions, evolving independently over the different layers across the thickness of the device through N−1 Allen-Cahn type evolution equations per layer. A comprehensive series of numerical examples is analysed, with the aim of exploring the capability of the proposed methodology to propose efficient optimal designs. Specifically, the topology optimisation algorithm determines the topology of regions where different DE materials must be conveniently placed in order to attain complex electrically induced configurations.

Low regularity integrators for semilinear parabolic equations with maximum bound principles

This paper is concerned with conditionally structure-preserving, low regularity time integration methods for a class of semilinear parabolic equations of Allen–Cahn type. Important properties of such equations include maximum bound principle (MBP) and energy dissipation law; for the former, that means the absolute value of the solution is pointwisely bounded for all the time by some constant imposed by appropriate initial and boundary conditions. The model equation is first discretized in space by the central finite difference, then by iteratively using Duhamel’s formula, first- and second-order low regularity integrators (LRIs) are constructed for time discretization of the semi-discrete system. The proposed LRI schemes are proved to preserve the MBP and the energy stability in the discrete sense. Furthermore, their temporal error estimates are also successfully derived under a low regularity requirement that the exact solution of the semi-discrete problem is only assumed to be continuous in time. Numerical results show that the proposed LRI schemes are more accurate and have better convergence rates than classic exponential time differencing schemes, especially when the interfacial parameter approaches zero.

A second-order unconditionally energy stable scheme for phase-field based multimaterial topology optimization

A multimaterial topology optimization problem is solved by introducing a numerical scheme with fast convergence, second-order accuracy and unconditional energy stability. The modeling is based on the energy of multi-phase-field elasticity system including the classical Ginzburg–Landau, the elastic potential and some constraints. The material layout is updated by using the volume constrained gradient flow of the system. In this work, we transform the traditional objective functional of multimaterial topology optimization into the energy functional of multi-phase-field elasticity system, and transform the optimal material layout into the solutions of volume constrained Allen–Cahn type equations. For these Allen–Cahn type equations, we propose the second-order unconditionally energy stable numerical scheme which combines linearly stabilized splitting method and Crank–Nicolson scheme. For the proposed second-order scheme, we give a theoretical proof of unconditional energy stability. Numerical results show that the scheme converges fast compared to traditional Cahn–Hilliard type equations. Some classical benchmarks are performed to verify the feasibility and efficiency of our method.

Asymptotic behavior of an Allen-Cahn type equation with temperature

In this paper, we are interested in the study of the asymptotic behavior of an Allen-Cahn type equation with temperature and endowed Dirichlet boundary conditions. Such a model has, in particular, applications in chemistry. In particular, we obtain well-posedness results and the existence of the finite-dimensional global attractor as well as the existence of exponentials attractors by means of classical arguments.