Allen-Cahn Equation Research Articles

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

The critical variational setting for stochastic evolution equations

In this paper we introduce the critical variational setting for parabolic stochastic evolution equations of quasi- or semi-linear type. Our results improve many of the abstract results in the classical variational setting. In particular, we are able to replace the usual weak or local monotonicity condition by a more flexible local Lipschitz condition. Moreover, the usual growth conditions on the multiplicative noise are weakened considerably. Our new setting provides general conditions under which local and global existence and uniqueness hold. In addition, we prove continuous dependence on the initial data. We show that many classical SPDEs, which could not be covered by the classical variational setting, do fit in the critical variational setting. In particular, this is the case for the Cahn–Hilliard equation, tamed Navier–Stokes equations, and Allen–Cahn equation.

Bernoulli factories and duality in Wright–Fisher and Allen–Cahn models of population genetics

Mathematical models of genetic evolution often come in pairs, connected by a so-called duality relation. The most seminal example are the Wright–Fisher diffusion and the Kingman coalescent, where the former describes the stochastic evolution of neutral allele frequencies in a large population forwards in time, and the latter describes the genetic ancestry of randomly sampled individuals from the population backwards in time. As well as providing a richer description than either model in isolation, duality often yields equations satisfied by quantities of interest. We employ the so-called Bernoulli factory – a celebrated tool in simulation-based computing – to derive duality relations for broad classes of genetics models. As concrete examples, we present Wright–Fisher diffusions with general drift functions, and Allen–Cahn equations with general, nonlinear forcing terms. The drift and forcing functions can be interpreted as the action of frequency-dependent selection. To our knowledge, this work is the first time a connection has been drawn between Bernoulli factories and duality in models of population genetics.

Quantitative convergence of the vectorial Allen–Cahn equation towards multiphase mean curvature flow

Phase-field models such as the Allen–Cahn equation may give rise to the formation and evolution of geometric shapes, a phenomenon that may be analyzed rigorously in suitable scaling regimes. In its sharp-interface limit, the vectorial Allen–Cahn equation with a potential with N\geq 3 distinct minima has been conjectured to describe the evolution of branched interfaces by multiphase mean curvature flow. In the present work, we give a rigorous proof for this statement in two and three ambient dimensions and for a suitable class of potentials: as long as a strong solution to multiphase mean curvature flow exists, solutions to the vectorial Allen–Cahn equation with well-prepared initial data converge towards multiphase mean curvature flow in the limit of vanishing interface width parameter \varepsilon\searrow 0 . We even establish the rate of convergence O(\varepsilon^{1/2}) . Our approach is based on the gradient-flow structure of the Allen–Cahn equation and its limiting motion: building on the recent concept of “gradient-flow calibrations” for multiphase mean curvature flow, we introduce a notion of relative entropy for the vectorial Allen–Cahn equation with multi-well potential. This enables us to overcome the limitations of other approaches, e.g. avoiding the need for a stability analysis of the Allen–Cahn operator or additional convergence hypotheses for the energy at positive times.

TCAS-PINN: Physics-informed neural networks with a novel temporal causality-based adaptive sampling method

Physics-informed neural networks (PINNs) have become an attractive machine learning framework for obtaining solutions to partial differential equations (PDEs). PINNs embed initial, boundary, and PDE constraints into the loss function. The performance of PINNs is generally affected by both training and sampling. Specifically, training methods focus on how to overcome the training difficulties caused by the special PDE residual loss of PINNs, and sampling methods are concerned with the location and distribution of the sampling points upon which evaluations of PDE residual loss are accomplished. However, a common problem among these original PINNs is that they omit special temporal information utilization during the training or sampling stages when dealing with an important PDE category, namely, time-dependent PDEs, where temporal information plays a key role in the algorithms used. There is one method, called Causal PINN, that considers temporal causality at the training level but not special temporal utilization at the sampling level. Incorporating temporal knowledge into sampling remains to be studied. To fill this gap, we propose a novel temporal causality-based adaptive sampling method that dynamically determines the sampling ratio according to both PDE residual and temporal causality. By designing a sampling ratio determined by both residual loss and temporal causality to control the number and location of sampled points in each temporal sub-domain, we provide a practical solution by incorporating temporal information into sampling. Numerical experiments of several nonlinear time-dependent PDEs, including the Cahn–Hilliard, Korteweg–de Vries, Allen–Cahn and wave equations, show that our proposed sampling method can improve the performance. We demonstrate that using such a relatively simple sampling method can improve prediction performance by up to two orders of magnitude compared with the results from other methods, especially when points are limited.

Phase-field modeling and effective simulation of non-isothermal reactive transport

We consider single-phase flow with solute transport where ions in the fluid can precipitate and form a mineral, and where the mineral can dissolve and release solute into the fluid. Such a setting includes an evolving interface between fluid and mineral. We approximate the evolving interface with a diffuse interface, which is modeled with an Allen–Cahn equation. We also include effects from temperature such that the reaction rate can depend on temperature, and allow heat conduction through fluid and mineral. As Allen–Cahn is generally not conservative due to curvature-driven motion, we include a reformulation that is conservative. This reformulation includes a non-local term that makes the use of standard Newton iterations for solving the resulting non-linear system of equations very slow. We instead apply L-scheme iterations, which can be proven to converge for any starting guess, although giving only linear convergence. The three coupled equations for diffuse interface, solute transport and heat transport are solved via an iterative coupling scheme. This allows the three equations to be solved more efficiently compared to a monolithic scheme, and only few iterations are needed for high accuracy. Through numerical experiments we highlight the usefulness and efficiency of the suggested numerical scheme and the applicability of the resulting model.

A Unified MPM Framework Supporting Phase-field Models and Elastic-viscoplastic Phase Transition

Recent years have witnessed the rapid deployment of numerous physics-based modeling and simulation algorithms and techniques for fluids, solids, and their delicate coupling in computer animation. However, it still remains a challenging problem to model the complex elastic-viscoplastic behaviors during fluid–solid phase transitions and facilitate their seamless interactions inside the same framework. In this article, we propose a practical method capable of simulating granular flows, viscoplastic liquids, elastic-plastic solids, rigid bodies, and interacting with each other, to support novel phenomena all heavily involving realistic phase transitions, including dissolution, melting, cooling, expansion, shrinking, and so on. At the physics level, we propose to combine and morph von Mises with Drucker–Prager and Cam–Clay yield models to establish a unified phase-field-driven EVP model, capable of describing the behaviors of granular, elastic, plastic, viscous materials, liquid, non-Newtonian fluids, and their smooth evolution. At the numerical level, we derive the discretization form of Cahn–Hilliard and Allen–Cahn equations with the material point method to effectively track the phase-field evolution, so as to avoid explicit handling of the boundary conditions at the interface. At the application level, we design a novel heuristic strategy to control specialized behaviors via user-defined schemes, including chemical potential, density curve, and so on. We exhibit a set of numerous experimental results consisting of challenging scenarios to validate the effectiveness and versatility of the new unified approach. This flexible and highly stable framework, founded upon the unified treatment and seamless coupling among various phases, and effective numerical discretization, has its unique advantage in animation creation toward novel phenomena heavily involving phase transitions with artistic creativity and guidance.

A numerical study of interface dynamics in fluid materials

This paper deals with the approximation of the dynamics of two fluids having non-matching densities and viscosities. The modeling involves the coupling of the Allen-Cahn equation with the time-dependent Navier-Stokes equations. The Allen-Cahn equation describes the evolution of a scalar order parameter that assumes two distinct values in different spatial regions. Conversely, the Navier-Stokes equations govern the movement of a fluid subjected to various forces like pressure, gravity, and viscosity. When the Allen-Cahn equation is coupled with the Navier-Stokes equations, it is typically done through a surface tension term. The surface tension term accounts for the energy required to create an interface between the two phases, and it is proportional to the curvature of the interface. The Navier-Stokes equations are modified to include this term, which leads to the formation of a dynamic interface between the two phases. The resulting system of equations is known as the two-phase Navier-Stokes/Allen-Cahn equations. In this paper, the authors propose a mathematical model that combines the Allen-Cahn model and the Navier-Stokes equations to simulate multiple fluid flows. The Allen-Cahn model is utilized to represent the diffuse interface between different fluids, while the Navier-Stokes equations are employed to describe the fluid dynamics. The Allen-Cahn-Navier-Stokes model has been employed to simulate the generation of bubbles in a liquid subjected to an acoustic field. The model successfully predicted the size of the bubbles and the frequency at which they formed. The numerical outcomes were validated against experimental data, and a favorable agreement was observed.

Jet space extensions of infinite-dimensional Hamiltonian systems

We analyze infinite-dimensional Hamiltonian systems corresponding to partial Differential equations on one-dimensional spatial domains formulated with formally skew-adjoint Hamiltonian operators and non-quadratic Hamiltonian density. In various applications, the Hamiltonian density can depend on spatial derivatives of the state such that these systems can not straightforwardly be formulated as boundary port-Hamiltonian system using a Stokes-Dirac structure. In this work, we show that any Hamiltonian system of the above class can be reformulated as a Hamiltonian system on the jet space, in which the Hamiltonian density only depends on the extended state variable itself and not on its derivatives. Consequently, well-known geometric formulations with Stokes-Dirac structures are applicable. Additionally, we provide a similar result for dissipative systems. We illustrate the developed theory by means of the the Boussinesq equation, the dynamics of an elastic rod and the Allen-Cahn equation.

Phase-field simulation on fission gas release behavior of large grain UO<sub>2</sub> fuel

In order to predict the release behavior of fission gas in large grain UO<sub>2</sub> fuel and provide support for the development of accident tolerant fuel, a phase-field model is used to simulate the release behavior of fission gas in the microstructure of UO<sub>2</sub> polycrystalline in this work. This model adopts a set of coupled Cahn-Hilliard equations and Allen-Cahn equations, using conserved field variables to represent the distribution of fission gas and vacancies, and distinguishing bubble phase from matrix phase by using order parameters. This model focuses on investigating the effects of different grain sizes, temperature conditions, and diffusion coefficients on the release behavior of fission gas, demonstrating the nucleation, growth, and fusion behavior of bubbles. Simulation results are obtained for fuel porosity, bubble coverage on grain boundaries, and average bubble radius at a certain degree of burnup. The results show that temperature and diffusion coefficient have a significant influence on porosity and bubble coverage on grain boundaries. When the diffusion coefficient is high, grain size also has a significant influence on fission gas release behavior. And when the diffusion coefficient is low, the influence of grain size is not significant. In addition, the distribution of fission gas bubbles under high burnup obtained through this model is also in good agreement with experimental result. The model can predict the behavior of fission gas release in large grain UO<sub>2</sub> fuel.

Stability analysis of an explicit numerical scheme for the Allen-Cahn equation with high-order polynomial potentials

<abstract><p>The Allen-Cahn (AC) model is a mathematical equation that represents the phase separation process. The AC equation has numerous applications in various disciplines, such as image processing, physics, and biology. It models phase transitions, such as solidification and grain growth in materials, pattern formation in chemical reactions, and domain coarsening in biological systems like lipid membranes. Numerical methods are crucial for solving the AC equation due to its complexity and nonlinear nature. Analytical solutions are often extremely difficult to obtain. Therefore, the development of efficient numerical techniques is indispensable for approximating solutions and studying phase transitions, material behavior, and pattern formation accurately. We investigate the stability of an explicit finite difference method (FDM) used to numerically solve the two-dimensional (2D) AC model with a high-order polynomial potential, which was recently proposed to preserve a more intricate structure of interfaces. To demonstrate the precision and optimal estimate of our stability constraints, we conduct various computational tests using the derived time step formulas that ensure the maximum principle.</p></abstract>

An explicit fourth-order accurate compact method for the Allen-Cahn equation

<abstract><p>In this paper, we propose an explicit spatially fourth-order accurate compact scheme for the Allen-Cahn equation in one-, two-, and three-dimensional spaces. The proposed method is based on the explicit Euler time integration scheme and fourth-order compact finite difference method. The proposed numerical solution algorithm is highly efficient and simple to implement because it is an explicit scheme. There is no need to solve implicitly a system of discrete equations as in the case of implicit numerical schemes. Furthermore, when we consider the temporally accurate numerical solutions, the time step restriction is not severe because the governing equation is a second-order parabolic partial differential equation. Computational tests are conducted to demonstrate the superior performance of the proposed spatially fourth-order accurate compact method for the Allen-Cahn equation.</p></abstract>

An efficient technique based on barycentric interpolation collocation method for the time fractional Allen-Cahn equation

An efficient technique based on barycentric interpolation collocation method for the time fractional Allen-Cahn equation

Optimal Error Estimates of a Discontinuous Galerkin Method for Stochastic Allen-Cahn Equation Driven by Multiplicative Noise

Optimal Error Estimates of a Discontinuous Galerkin Method for Stochastic Allen-Cahn Equation Driven by Multiplicative Noise

Numerical modeling of stress corrosion cracking in steel structures with phase field method

This study uses the phase field method to present a coupled mechano-electro-chemical formulation for predicting stress corrosion cracking (SCC) phenomena in steel structures. SCC constitutes an intricate damage process stemming from the interplay between mechanical loading and corrosion within an aggressive environment. The formulation presented herein introduces a phase-field parameter that aggregates the mechanical and electrochemical influences. To achieve this goal, the internal energies governing the SCC phenomenon are separated into elastic-damage strain energy, interfacial reaction energy, and energy resulting from changes in corrosion ion concentration. The Allen-Cahn equation is modified to include all energy contributions and calculate the phase-field parameter. Furthermore, a specific interfacial kinetic coefficient is introduced to the mechanical energy to consider corrosion current effects on mechanical properties. The Cahn-Hilliard equation is applied to model the corrosion ion concentration in the domain, and the mechanical state of the body is obtained by solving the equilibrium equations. Several numerical examples are presented to indicate the robustness and accuracy of the proposed formulation. In this respect, the method is applied to predict crack propagation resulting from SCC in two practical engineering problems, yielding promising results.

A Hard-Constraint Wide-Body Physics-Informed Neural Network Model for Solving Multiple Cases in Forward Problems for Partial Differential Equations

In the fields of physics and engineering, it is crucial to understand phase transition dynamics. This field involves fundamental partial differential equations (PDEs) such as the Allen–Cahn, Burgers, and two-dimensional (2D) wave equations. In alloys, the evolution of the phase transition interface is described by the Allen–Cahn equation. Vibrational and wave phenomena during phase transitions are modeled using the Burgers and 2D wave equations. The combination of these equations gives comprehensive information about the dynamic behavior during a phase transition. Numerical modeling methods such as finite difference method (FDM), finite volume method (FVM) and finite element method (FEM) are often applied to solve phase transition problems that involve many partial differential equations (PDEs). However, physical problems can lead to computational complexity, increasing computational costs dramatically. Physics-informed neural networks (PINNs), as new neural network algorithms, can integrate physical law constraints with neural network algorithms to solve partial differential equations (PDEs), providing a new way to solve PDEs in addition to the traditional numerical modeling methods. In this paper, a hard-constraint wide-body PINN (HWPINN) model based on PINN is proposed. This model improves the effectiveness of the approximation by adding a wide-body structure to the approximation neural network part of the PINN architecture. A hard constraint is used in the physically driven part instead of the traditional practice of PINN constituting a residual network with boundary or initial conditions. The high accuracy of HWPINN for solving PDEs is verified through numerical experiments. One-dimensional (1D) Allen–Cahn, one-dimensional Burgers, and two-dimensional wave equation cases are set up for numerical experiments. The properties of the HWPINN model are inferred from the experimental data. The solution predicted by the model is compared with the FDM solution for evaluating the experimental error in the numerical experiments. HWPINN shows great potential for solving the PDE forward problem and provides a new approach for solving PDEs.

Solutions of the Ginzburg–Landau equations with vorticity concentrating near a nondegenerate geodesic

It is well-known that under suitable hypotheses, for a sequence of solutions of the (simplified) Ginzburg–Landau equations -\Delta u_{\varepsilon} +\varepsilon^{-2}(|u_{\varepsilon}|^{2}-1)u_{\varepsilon} = 0 , the energy and vorticity concentrate as \varepsilon\to 0 around a codimension 2 stationary varifold – a (measure-theoretic) minimal surface. Much less is known about the question of whether, given a codimension 2 minimal surface, there exists a sequence of solutions for which the given minimal surface is the limiting concentration set. The corresponding question is very well-understood for minimal hypersurfaces and the scalar Allen–Cahn equation, and for the Ginzburg–Landau equations when the minimal surface is locally area-minimizing, but otherwise quite open. We consider this question on a 3 -dimensional closed Riemannian manifold (M,g) , and we prove that any embedded nondegenerate closed geodesic can be realized as the asymptotic energy/ vorticity concentration set of a sequence of solutions of the Ginzburg–Landau equations.

Coupled mechano-electro-thermal model to predict phase transition in phase change materials

An advanced coupled Mechano-Electro-Thermal Phase Field model is presented to capture the crystallization/amorphization of phase change nanofilm under external voltage. It consisted of the Allen-Cahn equation for phase transition (PT), elasticity equations to resolve stresses, heat conduction, and the Poisson equation to consider thermoelectricity. The proposed model depicts the intricate interplay between temperature, stress, pre-existing defects (surface void), and PT in model phase change material, i.e., Germanium-antimony-tellurium (GeSbTe or GST). The predicted stress generation, thermal and PT-induced, agrees with existing experimental data. Additionally, surface voids can cause remarkable temperature and PT kinetics changes, decreasing the transformation rate. Also, due to a highly heterogeneous temperature, the thermal strain shows a crucial effect on crystallization kinetics. Finally, the critical stepwise voltage (Vcr) is calculated, representing an inverse power relation with the pulse time, i.e., the maximum voltage results in complete crystallization below the melting temperature. An initial increase in the average temperature occurs at the critical voltage, followed by a constant temperature regime as crystalization evolves and a jump to higher temperatures until complete crystallization. However, the temperature rapidly jumps above the melting temperature for larger voltages (V > Vcr), leading to the reverse transformation, i.e., amorphization.

Stability and Error Estimates of High Order BDF-LDG Discretizations for the Allen–Cahn Equation

We construct high order local discontinuous Galerkin (LDG) discretizations coupled with third and fourth order backward differentiation formulas (BDF) for the Allen–Cahn equation. The numerical discretizations capture the advantages of linearity and high order accuracy in both space and time. We analyze the stability and error estimates of the time third-order and fourth-order BDF-LDG discretizations for numerically solving Allen–Cahn equation respectively. Theoretical analysis shows the stability and the optimal error results of theses numerical discretizations, in the sense that the time step τ requires only a positive upper bound and is independent of the mesh size h. A series of numerical examples show the correctness of the theoretical analysis. Comparison with the first-order numerical discretization illustrates that the high order BDF-LDG discretizations show good performance in solving stiff problems.

Recovering critical parameter for nonlinear Allen–Cahn equation by fully discrete continuous data assimilation algorithms *

The purpose of this study is to recover the diffuse interface width parameter for nonlinear Allen–Cahn equation by a continuous data assimilation algorithm proposed recently. We obtain the large-time error between the true solution of the Allen–Cahn equation and the data assimilated solution produced by implicit–explicit one-leg fully discrete finite element methods due to discrepancy between an approximate diffuse interface width and the physical interface width. The strongly A-stability of the one-leg methods plays key roles in proving the exponential decay of initial error. Based on the long-time error estimates, we develop several algorithms to recover both the true solution and the true diffuse interface width using only spatially discrete phase field function measurements. Numerical experiments confirm our theoretical results and verify the effectiveness of the proposed methods.