Allen-Cahn Equation Research Articles

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

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

Maximum bound principle for matrix-valued Allen-Cahn equation and integrating factor Runge-Kutta method

Maximum bound principle for matrix-valued Allen-Cahn equation and integrating factor Runge-Kutta method

L-stable spectral deferred correction methods and applications to phase field models

This paper presents the L-stable spectral deferred correction (SDC) methods with low stages. These schemes are initiated by the Crank-Nicolson method. We adopt the linear stabilization approach for the phase field models to obtain the linear implicit SDC scheme. This is done by adding and subtracting the linear stabilization operators that are provided for the different phase field problems. Without loss of the low-stage property, the extrapolation technique is also used in the prediction step of the semi-implicit SDC method. Numerical experiments are given to validate the high-order accuracy and the energy decay property of the proposed semi-implicit SDC methods for the Allen-Cahn, Cahn-Hilliard, and molecular beam epitaxy equations.

A class of unconditionally energy stable relaxation schemes for gradient flows

In this paper, a novel class of unconditionally energy stable schemes are constructed for solving gradient flow models by combining the relaxed scalar auxiliary variable (SAV) approach with the linear multistep technique. The main idea is first to rewrite the original model as an equivalent system, and then the linear multistep method in time, together with the relaxation strategy is employed to compute the solution of the reformulated system. The main benefit of the proposed schemes is that it can achieve second-order temporal accuracy and strictly unconditional energy stability, as well as a quite close to, and in many cases equal to, the original energy than the baseline SAV approaches. Finally, ample numerical experiments for the Allen–Cahn equation and Cahn–Hilliard equation are presented to confirm the efficiency, accuracy and strictly unconditional energy stability of the proposed schemes.

Energy-dissipative evolutionary deep operator neural networks

Energy-Dissipative Evolutionary Deep Operator Neural Network is an operator learning neural network. It is designed to seek numerical solutions for a class of partial differential equations instead of a single partial differential equation, such as partial differential equations with different parameters or different initial conditions. The network consists of two sub-networks, the Branch net and the Trunk net. For an objective operator G, the Branch net encodes different input functions f at a fixed number of sensors xi,i=1,2,⋯,m, and the Trunk net evaluates the output function at any location. By minimizing the error between the evaluated output q and the expected output G(f)(y) at the test point y, DeepONet generates a good approximation of the operator G. A key distinction in our methodology is the utilization of DeepONet for the training of the initial state, which operates as a multi-parametric operator. Further, the evolution of parameters in our model is facilitated by the Scalar Auxiliary Variable (SAV) method, leading to a formulated iterative process for the parameter's progression over time. The SAV approach is adopted to preserve essential physical properties of PDEs, particularly the Energy Dissipation Law. It introduces a kind of modified energy and establishes unconditional energy dissipation law in the discrete level. By treating the parameters of the well-trained DeepONet as a representation of the initial operator and evolving them by a dynamic system, our network can predict the accurate solution at any further time, while the training data is only the initial state. In order to validate the accuracy and efficiency of our neural networks, we provide numerical simulations of several partial differential equations, including heat equations, parametric heat equations, Allen-Cahn equations and a reaction–diffusion equation in three dimensions.

Second-Order Estimates for Transition Layers and a Curvature Estimate for the Parabolic Allen–Cahn

Abstract The parabolic Allen–Cahn equation is a semilinear partial differential equation that is closely linked to the mean curvature flow by a singular perturbation. Motivated by the work of Wang–Wei [ 21] and Chodosh–Mantoulidis [ 3] in the elliptic setting, we initiate the corresponding regularity theory for parabolic Allen–Cahn flows. In particular, we establish an improved convergence property of parabolic Allen–Cahn flows to the mean curvature flow: if the phase-transition level sets converge in $C^{2}$, then they converge in $C^{2,\theta }$ as well. As an application, we obtain a curvature estimate for the parabolic Allen–Cahn equation, which can be seen as a diffused version of Brakke’s [ 1] and White’s [ 24] regularity theorems for mean curvature flow.

Multioutput FOSLS Deep Neural Network for Solving Allen–Cahn Equation

Multioutput FOSLS Deep Neural Network for Solving Allen–Cahn Equation

Diffusion in Allen–Cahn equation: Normal vs anomalous

It is known that the anomalous diffusion can be well characterized by the fractional derivatives. In this paper, the Allen–Cahn equations with different kinds of time fractional derivatives are numerically studied. The numerical solutions of the Allen–Cahn equations are obtained by using Euler method or L1 method to approximate the time derivative and the finite element method in the space direction. Finally, the influences of time derivatives on the solutions of the considered models are observed and discussed.

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.

Investigation on interphase mass transfer coefficient of a deformable bubble with density contrast using phase-field lattice Boltzmann method

The gas–liquid mass transfer process with density contrast is modeled in this paper using a phase-field Lattice Boltzmann (LB) method. In this method, the phase-field model used to calculate the flow field and the continuous species transfer model used to calculate the concentration distribution are coupled by introducing the divergence condition of the velocity in the presence of mass transfer into the Allen-Cahn equation. For a deformable bubble rising in stagnant liquid, a two-dimensional parameter dependence study is conducted and the results indicate that an increase in initial diameter, as well as a decrease in liquid diffusion coefficient or liquid viscosity, can improve the steady-state value for the Sherwood number, whereas the distribution coefficient has a little impact on this steady-state value. Furthermore, a correlation originally proposed by Takemura and Yabe (1998) modified with a shape factor can be used to predict the steady-state Sherwood number in the numerical results.

On the construction of an efficient finite-element solver for phase-field simulations of many-particle solid-state-sintering processes

We present an efficient solver for the simulation of many-particle solid-state-sintering processes. The microstructure evolution is described by a system of equations consisting of one Cahn–Hilliard equation and a set of Allen-Cahn equations to distinguish neighboring particles. The particle packing is discretized in space via multicomponent linear adaptive finite elements and implicitly in time with variable time-step sizes, resulting in a large nonlinear system of equations with strong coupling between all components to be solved. Since on average 10k degrees of freedom per particle are necessary to accurately capture the interface dynamics in 3D, we propose strategies to solve the resulting large and challenging systems. This includes the efficient evaluation of the Jacobian matrix as well as the implementation of Jacobian-free methods by applying state-of-the-art matrix-free algorithms for high and dynamic numbers of components, advances regarding preconditioning, and a fully distributed grain-tracking algorithm. We validate the obtained results, examine in detail the node-level performance and demonstrate the scalability up to 10k particles on modern supercomputers. Such numbers of particles are sufficient to simulate the sintering process in (statistically meaningful) representative volume elements. Our framework thus forms a valuable tool for the virtual design of solid-state-sintering processes for pure metals and their alloys.

The adaptive SAV weak Galerkin finite element method for the Allen-Cahn equation

In this paper, the weak Galerkin finite element method with the scalar auxiliary variable (SAV) approach is considered for the Allen-Cahn equation. Based on the elliptic reconstruction technique, the elliptic equation corresponding to the Allen-Cahn equation is introduced, which is employed to split the numerical error into the elliptic error and the parabolic error. Then the weak gradient recovery type a posteriori error estimator of the elliptic equation is adopted to develop the time-space adaptive algorithm. The effectiveness of the SAV weak Galerkin finite element method and the time-space adaptive algorithm is verified by several numerical benchmarks on both uniform and adaptive meshes.

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.

Reinforcement learning strategy for control of microstructure evolution in phase field models

We propose a reinforcement learning-based approach for controlling microstructure evolution in phase field models. A key feature of our approach is the replacement of the high-fidelity but computationally expensive phase field simulation with an inexpensive surrogate model of acceptable fidelity. We apply our approach to a 2-D model problem based on the Allen-Cahn equation that represents a two-phase microstructure with a single, scalar order parameter. The control parameters for this model are spatially and temporally varying temperature (T) and bias fields (h). Our approach generates sets of T and h parameters that yield final microstructures qualitatively matching target microstructures. We discuss further developments required to apply this approach to optimal processing of designer microstructures.

Two-grid algorithm of lumped mass finite element approximation for Allen-Cahn equations

In this paper, we present a two-grid lumped mass finite element algorithm for 2D Allen-Cahn equations, where Crank-Nicolson scheme and piecewise linear element are utilized for temporal and spatial discretization, respectively. Both the maximum-norm boundedness and H1-norm error estimates of the proposed two-grid scheme are discussed. Finally, a numerical example is given to verify the theoretical results. By comparing with the finite element scheme and the two-grid finite element scheme, we found that our scheme has a great advantage in calculation time.

Collocation-based numerical simulation of fractional order Allen–Cahn equation

Collocation-based numerical simulation of fractional order Allen–Cahn equation

Discovering stochastic partial differential equations from limited data using variational Bayes inference

We propose a novel framework for discovering Stochastic Partial Differential Equations (SPDEs) from data. The proposed approach combines the concepts of stochastic calculus, variational Bayes theory, and sparse learning. We propose the extended Kramers–Moyal expansion to express the drift and diffusion terms of an SPDE in terms of state responses and use Spike-and-Slab priors with sparse learning techniques to efficiently and accurately discover the underlying SPDEs. The proposed approach has been applied to three canonical SPDEs, (a) stochastic heat equation, (b) stochastic Allen–Cahn equation, and (c) stochastic Nagumo equation. Our results demonstrate that the proposed approach can accurately identify the underlying SPDEs with limited data. This is the first attempt at discovering SPDEs from data, and it has significant implications for various scientific applications, such as climate modeling, financial forecasting, and chemical kinetics.

Framework for a chemo‐mechanical model of the breathing effect in lithium–silicon batteries

AbstractWe present a chemo‐mechanical finite element method model for the breathing effect of the silicon anode during lithiation and delithiation. The model is based on the framework for gradient extended standard dissipative solids. It couples diffusion of lithium ions, phase transformation with a moving phase boundary, and the related volume changes, which lead to mechanical stresses in the material. The driving force for diffusion is given by the gradient of the electrochemical potential. Phase transformation and phase boundary movement are described by the Allen–Cahn equation in combination with a constrained double obstacle potential. To shift the constraint from the nodes to the Gauss points, the model makes use of the micromorphic approach.

A Relation of the Allen–Cahn equations and the Euler equations and applications of the equipartition

We will prove that solutions of the Allen–Cahn equations that satisfy the equipartition of the energy can be transformed into solutions of the Euler equations with constant pressure. As a consequence, we obtain De Giorgi type results, that is, the level sets of entire solutions are hyperplanes. Also, we will determine the structure of solutions of the Allen–Cahn system in two dimensions that satisfy the equipartition. In addition, we apply the Leray projection on the Allen–Cahn system and provide some explicit entire solutions. Finally, we obtain some examples of smooth entire solutions of the Euler equations. For specific type of initial conditions, some of these solutions can be extended to the Navier–Stokes equations. The motivation of this paper is to find a transformation that relates the solutions of the Allen–Cahn equations to solutions of the minimal surface equation of one dimension less. We prove this result for equipartitioned solutions in dimension three.

SONets: Sub-operator learning enhanced neural networks for solving parametric partial differential equations

Complex nonlinear partial differential equations (PDEs) can be decomposed into subsystems comprising certain linear and nonlinear sub-operators. We usually have fundamental solutions to the linear subsystems and have a good knowledge of their uniqueness and regularity. The nonlinear subsystems have more straightforward formulas than the original complex nonlinear PDEs, even with reduced dimensions. Solutions of the complex nonlinear PDEs based on simpler subsystems are investigated in this study. We derive a proper subsystem decomposition of the nonlinear PDEs into linear and nonlinear subsystems to get unique solutions. We propose the sub-operator learning enhanced neural networks (SONets) based on this proper subsystem decomposition to solve nonlinear PDEs. The solutions of nonlinear subsystems with generated force terms are modeled via operator learning. In contrast, only the cumbersome integration and series parts of fundamental solutions of linear subsystems are modeled. The complex nonlinear PDEs are then solved by finding the force terms to minimize the physics-informed loss functions. It is also convenient to generalize SONets for a new PDE parameter, e.g., the diffusion coefficient. Next, we extend the SONets to a meta version (meta-SONets) for parametric PDEs by transfer learning. In meta-SONets, the initial conditions are explicitly encoded via operator learning neural networks (ENNs) to output force terms, followed by the whole SONets to solve PDEs. In the pre-training stage of meta-SONets, parametric PDEs given randomly generated initial conditions are solved by training ENNs to minimize the physics-informed loss functions. At the fine-tuning stage, the trainable parameters of ENNs are fine-tuned to solve the PDE given a new initial condition. Numerical experiments are conducted to solve the Burgers equation, the nonlinear diffusion-reaction system, and the Allen-Cahn equation. The numerical results indicate that SONets with neural network-parameterized force terms obtain solutions with good accuracy. With transfer learning, the computational cost of meta-SONets for solving the nonlinear PDEs with a new initial condition is dramatically reduced.