Allen-Cahn Type Equation Research Articles

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.

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.

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.

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.

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.

A second-order unconditionally stable method for the anisotropic dendritic crystal growth model with an orientation-field

In this article, we develop a linear, unconditionally energy stable computational scheme for solving the dendritic crystal growth model with the orientational field. We apply the phase field model to describe the evolution of crystal with rotation. The model, which couples the heat equation and anisotropic Allen–Cahn type equation, is a complicated nonlinear system. The time integration is based on the second-order Crank–Nicolson method. The anisotropic coefficient is treated by using the invariant energy quadratization. We mathematically prove that the proposed method is unconditionally energy stable. The second-order spatial and temporal accuracy will be preserved for the numerical approximation. Various computational tests are performed to show the accuracy, stability, and efficiency of the proposed scheme.

Stabilized Exponential-SAV Schemes Preserving Energy Dissipation Law and Maximum Bound Principle for The Allen–Cahn Type Equations

It is well-known that the Allen–Cahn equation not only satisfies the energy dissipation law but also possesses the maximum bound principle (MBP) in the sense that the absolute value of its solution is pointwise bounded for all time by some specific constant under appropriate initial/boundary conditions. In recent years, the scalar auxiliary variable (SAV) method and many of its variants have attracted much attention in numerical solutions for gradient flow problems due to their inherent advantage of preserving certain discrete analogues of the energy dissipation law. However, existing SAV schemes usually fail to preserve the MBP when applied to the Allen–Cahn equation. In this paper, we develop and analyze new first- and second-order stabilized exponential-SAV schemes for a class of Allen–Cahn type equations, which are shown to simultaneously preserve the energy dissipation law and MBP in discrete settings. In addition, optimal error estimates for the numerical solutions are rigorously obtained for both schemes. Extensive numerical tests and comparisons are also conducted to demonstrate the performance of the proposed schemes.

A phase field-based systematic multiscale topology optimization method for porous structures design

In this paper, we propose a novel and systematic phase field-based multiscale topology optimization method for porous structures design. The multi-regional microstructural composite and fixed microstructural shapes are adopted to balance the objective and other constraints. The connectivity issue between different microstructures is handled. In the multiscale topology optimization, the macro and micro design variables are updated by the Allen-Cahn type equations which include a reaction-diffusion term, a sensitivity analysis term, a volume constraint term, and a correction term. The correction term is used to keep topologies of macro and micro structures closed to the prescribed structures. We use an efficient merging algorithm to handle the connectivity issue between different microstructures. Based on an interpolation technique of the phase field function and a modified Allen-Cahn equation, this algorithm can smooth the connecting boundaries and satisfy the minimum surface theorem. The final distribution of microstructures keeps the extraordinary physical and mechanical properties for the macrostructure. Some typical cantilever beam, Michell-type structure and Messerschmitt-Bölkow-Blohm beam are performed to verify the effectiveness of our method.

Traveling wave dynamics for Allen-Cahn equations with strong irreversibility

Constrained gradient flows are studied in fracture mechanics to describestrongly irreversible(orunidirectional) evolution of cracks. The present paper is devoted to a study on the long-time behavior of non-compact orbits of such constrained gradient flows. More precisely, traveling wave dynamics for a one-dimensional fully nonlinear Allen-Cahn type equation involving the positive-part function is considered. Main results of the paper consist of a construction of a one-parameter family ofdegeneratetraveling wave solutions (even identified when coinciding up to translation) and exponential stability of such traveling wave solutions with some basin of attraction, although they are unstable in a usual sense.

Феноменологический вывод термомеханической модели развития канала электрического пробоя типа «диффузной границы»

In this work we derive diffuse-interface type model for electric breakdown evolution in solid dielectrics which accounts for non-isothermal and mechanical effects. The proposed model consists of mass, momentum and energy conservation equation, Maxwell’s equations in quasi(electro)static approximation and Allen-Cahn type equation which describes phase-field evolution. The derivation of the model is based on the rational thermomechanics framework, M. Gurtin’s microforce and microstress theory and Coleman-Noll procedure.

Explicit Hybrid Numerical Method for the Allen-Cahn Type Equations on Curved Surfaces

Explicit Hybrid Numerical Method for the Allen-Cahn Type Equations on Curved Surfaces

First and second order unconditionally energy stable schemes for topology optimization based on phase field method

In this paper, we use the phase field method to deal with the compliance minimization problem in topology optimization. A modified Allen-Cahn type equation with two penalty terms is proposed. The equation couples the diffusive interface dynamics and the linear elasticity mechanics. We propose the first- and second-order unconditionally energy stable schemes for the evolution of phase field modeling. The linearly stabilized splitting scheme is applied to improve the stability. The Crank-Nicolson scheme is applied to achieve second-order accuracy in time. We prove the unconditional stabilities of our schemes in analysis. The finite element method and the projected conjugate gradient method combining with fast fourier transform are used to solve the compliance minimization problem. Several experimental results are presented to verify the efficiency and accuracy of the proposed schemes.

Efficient, second-order in time, and energy stable scheme for a new hydrodynamically coupled three components volume-conserved Allen–Cahn phase-field model

In this paper, we establish a new hydrodynamically coupled phase-field model for three immiscible fluid components system. The model consists of the Navier–Stokes equations and three coupled nonlinear Allen–Cahn type equations, to which we add nonlocal type Lagrange multipliers to conserve the volume of each phase accurately. To solve the model, a linear and energy stable time-marching method is constructed by combining the stabilized-Invariant Energy Quadratization (S-IEQ) approach and the projection method. The well-posedness of the scheme and its unconditional energy stability are rigorously proved. Several numerical simulations in 2D and 3D are carried out, including spinodal decomposition, dynamical deformations of a liquid lens and rising liquid drops, to validate the model and demonstrate the efficiency and energy stability of the proposed scheme, numerically.

Solving Allen-Cahn and Cahn-Hilliard Equations Using the Adaptive Physics Informed Neural Networks

Phase field models, in particular, the Allen-Cahn type and Cahn-Hilliard type equations, have been widely used to investigate interfacial dynamic problems. Designing accurate, efficient, and stable numerical algorithms for solving the phase field models has been an active field for decades. In this paper, we focus on using the deep neural network to design an automatic numerical solver for the Allen-Cahn and Cahn-Hilliard equations by proposing an improved physics informed neural network (PINN). Though the PINN has been embraced to investigate many differential equation problems, we find a direct application of the PINN in solving phase-field equations won't provide accurate solutions in many cases. Thus, we propose various techniques that add to the approximation power of the PINN. As a major contribution of this paper, we propose to embrace the adaptive idea in both space and time and introduce various sampling strategies, such that we are able to improve the efficiency and accuracy of the PINN on solving phase field equations. In addition, the improved PINN has no restriction on the explicit form of the PDEs, making it applicable to a wider class of PDE problems, and shedding light on numerical approximations of other PDEs in general.

A level set approach for multi-layered interface systems

In this paper, a multi-layered interface system is introduced, which is formally derived by a singular limit of a weakly coupled system of the Allen–Cahn type equation. By using the level set approach, this system is written as a quasi-monotone degenerate parabolic system with discontinuous functions. The well-posedness of viscosity solutions is shown, and the singularity of particular viscosity solutions is discussed.

A new interface capturing method for Allen-Cahn type equations based on a flow dynamic approach in Lagrangian coordinates, I. One-dimensional case

We develop a new Lagrangian approach — flow dynamic approach to effectively capture the interface in the Allen-Cahn type equations. The underlying principle of this approach is the Energetic Variational Approach (EnVarA), motivated by Rayleigh and Onsager [27,28]. Its main advantage, comparing with numerical methods in Eulerian coordinates, is that thin interfaces can be effectively captured with few points in the Lagrangian coordinate. We concentrate in the one-dimensional case and construct numerical schemes for the trajectory equation in Lagrangian coordinate that obey the variational structures, and as a consequence, are energy dissipative. Ample numerical results are provided to show that only fewer points are enough to resolve very thin interfaces by using our flow dynamic approach.

Efficient numerical scheme for a new hydrodynamically-coupled conserved Allen–Cahn type Ohta–Kawaski phase-field model for diblock copolymer melt

In this work, we first propose a new hydrodynamically-coupled phase-field model for the diblock copolymer melt that is derived based on the Allen–Cahn dynamics where the volume fraction of two composing monomers is conserved by using a nonlocal Lagrange multiplier. Formally, the system is a highly nonlinear system that consists of the incompressible Navier–Stokes equations and a nonlocal-type Allen–Cahn type equation. Then, to solve the model, we develop a linear and second-order time-marching scheme via the Invariant Energy Quadratization approach with the stabilization technique for the nonlinear potentials, as well as the projection method for the Navier–Stokes equations. The unconditional energy stability of the numerical method is proved, and several experiments of 2D and 3D are performed to validate the accuracy and energy stability of the developed numerical scheme.

An efficient explicit full-discrete scheme for strong approximation of stochastic Allen–Cahn equation

In Becker and Jentzen (2019) and Becker et al. (2017), an explicit temporal semi-discretization scheme and a space–time full-discretization scheme were, respectively, introduced and analyzed for the additive noise-driven stochastic Allen–Cahn type equations, with strong convergence rates recovered. The present work aims to propose a different explicit full-discrete scheme to numerically solve the stochastic Allen–Cahn equation with cubic nonlinearity, perturbed by additive space–time white noise. The approximation is easily implementable, performing the spatial discretization by a spectral Galerkin method and the temporal discretization by a kind of nonlinearity-tamed accelerated exponential integrator scheme. Error bounds in a strong sense are analyzed for both the spatial semi-discretization and the spatio-temporal full discretization, with convergence rates in both space and time explicitly identified. It turns out that the obtained convergence rate of the new scheme is, in the temporal direction, twice as high as existing ones in the literature. Numerical results are finally reported to confirm the previous theoretical findings.