Modified Allen-Cahn Equation Research Articles

A fast, efficient, and explicit phase-field model for 3D mesh denoising

In this paper, we propose a fast and efficient explicit three-dimensional (3D) mesh denoising algorithm that utilizes the Allen–Cahn (AC) equation with a fidelity term. The phase-field model is used to describe the characteristics of both the surface and interior of an object, allowing us to represent the 3D mesh model with noise using a phase-field function. By using the phase separation property of the AC equation and the fidelity term, the model can effectively preserve the original structures and features during the smoothing process, even in the presence of noise in various regions. The modified AC equation is numerically discretized using the explicit finite difference method, where the values at neighboring grid points are used as Dirichlet boundary conditions. Because the algorithm is local and explicit, it guarantees both effective denoising of 3D mesh models and rapid implementation speed. To validate the efficacy of the proposed algorithm, we conduct various computational experiments. Furthermore, we propose an implicit-explicit numerical scheme using the Crank–Nicolson method to address the denoising problem of 3D mesh models and perform related experiments.

Lattice Boltzmann method for interface capturing.

Accurately solving phase interface plays a great role in modeling an immiscible multiphase flow system. In this paper, we propose an accurate interface-capturing lattice Boltzmann method from the perspective of the modified Allen-Cahn equation(ACE). The modified ACE is built based on the commonly used conservative formulation via the relation between the signed-distance function and the order parameter also maintaining the mass-conserved characteristic. A suitable forcing term is carefully incorporated into the lattice Boltzmann equationfor correctly recovering the target equation. We then test the proposed method by simulating some typical interface-tracking problems of Zalesaks disk rotation, single vortex, deformation field and demonstrate that the present model can be more numerically accurate than the existing lattice Boltzmann models for the conservative ACE, especially at a small interface-thickness scale.

Assessment of an energy-based surface tension model for simulation of two-phase flows using second-order phase field methods

Second-order phase field models have emerged as an attractive option for capturing the advection of interfaces in two-phase flows. Prior to these, state-of-the-art models based on the Cahn-Hilliard equation, which is a fourth-order equation, allowed for the derivation of surface tension models through thermodynamic arguments. In contrast, the second-order phase field models do not follow a known energy law, and deriving a surface tension term for these models using thermodynamic arguments is not straightforward. In this work, we justify that the energy-based surface tension model from the Cahn-Hilliard context can be adopted for second-order phase field models as well and assess its performance. We test the surface tension model on three different second-order phase field equations; the conservative diffuse interface model of Chiu and Lin [1], and two models based on the modified Allen-Cahn equation introduced by Sun and Beckermann [2]. Additionally, we draw the connection between the energy-based model with a localized variation of the continuum surface force (CSF) model. Using canonical tests, we illustrate the lower magnitude of spurious currents, better accuracy, and superior convergence properties of the energy-based surface tension model compared to the CSF model, which is a popular choice used in conjunction with second-order phase field methods, and the localized CSF model. Importantly, in terms of computational expense and parallel efficiency, the energy-based model incurs no penalty compared to the CSF models.

Surface reconstruction algorithm using a modified Allen–Cahn equation

In this paper, we propose a novel efficient surface reconstruction method from unorganized point cloud data in three-dimensional Euclidean space. The proposed method is based on the Allen–Cahn partial differential equation, with an edge indicating function to restrict the evolution. We applied the explicit Euler’s method to solve the discrete equation, and use the operator splitting technique to split the governing equation. Furthermore, we also modify the double well form to a periodic potential. Then we find that the proposed model can reconstruct the surface well even in the case of insufficient data. After selecting the appropriate parameters, we carried out various numerical experiments to demonstrate the robustness and accuracy of the proposed method. We adopt the proposed method to reconstruct the surfaces on simple, irregular and complex models, respectively, and can obtain smooth three-dimensional surfaces and visual effects. In addition, we also perform comparison tests to show the superiority of the proposed model. Statistic metrics such as the [Formula: see text], [Formula: see text], [Formula: see text], CPU time, and vertex numbers are evaluated. Results show that our model performs better than the other methods in statistical metrics even use far less point cloud data, and with the faster CPU computing speed.

Weighted 3D volume reconstruction from series of slice data using a modified Allen–Cahn equation

In this study, we develop a fast and accurate computational method for a weighted three-dimensional (3D) volume reconstruction from a series of slice data using a phase-field model. The proposed method is based on a modified Allen–Cahn (AC) equation with a fidelity term. The algorithm automatically generates the necessary slices between the given slices by solving the governing equation. To reconstruct a 3D volume, we first set a source slice and target slice. Next, we set the source slice as the initial condition and the target slice as the fidelity function. Finally, we retain the numerical solutions during an evolution as intermediate slices between the source and target slices. There are two criteria for choosing the intermediate slice: One is based on the area of the symmetric difference between the phase-field solution and the target and the other is based on the change of the phase-field solution relative to the area of the target. We use the weighted average of the two criteria. To validate the efficiency and accuracy of the proposed numerical algorithm, several computational experiments are conducted. Computational test results confirm the superior performance of the proposed algorithm.

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.

Motion by Mean Curvature with Constraints Using a Modified Allen–Cahn Equation

Motion by Mean Curvature with Constraints Using a Modified Allen–Cahn Equation

Investigation of bubble dynamics in a micro-channel with obstacles using a conservative phase-field lattice Boltzmann method

Simulating bubble dynamics impacting on obstacles is challenging because of large liquid-to-gas density ratio and complex interface deformation. In this study, a conservative phase-field model, based on a modified Allen–Cahn equation, is employed to accurately capture the bubble interface, and the lattice Boltzmann model is applied to solve the flow field. The bubble rises under the influence of buoyancy force and surface tension force, and complex topology changes, such as rotation, breakup, and squeeze deformation, are predicted in the presence of obstacles. Three dimensionless numbers, including Reynolds, Eötvös, and Morton numbers, are used to characterize bubble dynamics, and two shape indicators, including the revised Blaschke coefficient and the oblateness degree, are introduced to obtain a more systematic assessment of the bubble shape. Effects of flow parameters and obstacle geometries on bubble dynamics impacting on obstacles are investigated to render a quantitative investigation with physical insights. Model extension to the 3D case, the low-viscosity flow and non-pure fluid is further remarked, which can shed light onto future development of physically informed models for predicting the bubble behavior in more real scenarios.

Unconditionally stable exponential time differencing schemes for the mass‐conserving Allen–Cahn equation with nonlocal and local effects

AbstractIt is well known that the classic Allen–Cahn equation satisfies the maximum bound principle (MBP), that is, the absolute value of its solution is uniformly bounded for all time by certain constant under suitable initial and boundary conditions. In this paper, we consider numerical solutions of the modified Allen–Cahn equation with a Lagrange multiplier of nonlocal and local effects, which not only shares the same MBP as the original Allen–Cahn equation but also conserves the mass exactly. We reformulate the model equation with a linear stabilizing technique, then construct first‐ and second‐order exponential time differencing schemes for its time integration. We prove the unconditional MBP preservation and mass conservation of the proposed schemes in the time discrete sense and derive their error estimates under some regularity assumptions. Various numerical experiments in two and three dimensions are also conducted to verify the theoretical results.

Unconditionally Maximum Bound Principle Preserving Linear Schemes for the Conservative Allen–Cahn Equation with Nonlocal Constraint

In comparison with the Cahn–Hilliard equation, the classic Allen-Cahn equation satisfies the maximum bound principle (MBP) but fails to conserve the mass along the time. In this paper, we consider the MBP and corresponding numerical schemes for the modified Allen–Cahn equation, which is formed by introducing a nonlocal Lagrange multiplier term to enforce the mass conservation. We first study sufficient conditions on the nonlinear potentials under which the MBP holds and provide some concrete examples of nonlinear functions. Then we propose first and second order stabilized exponential time differencing schemes for time integration, which are linear schemes and unconditionally preserve the MBP in the time discrete level. Convergence of these schemes is analyzed as well as their energy stability. Various two and three dimensional numerical experiments are also carried out to validate the theoretical results and demonstrate the performance of the proposed schemes.

Automatic Binary Data Classification Using a Modified Allen–Cahn Equation

In this paper, we propose an automatic binary data classification method using a modified Allen–Cahn (AC) equation. The modified AC equation was originally developed for image segmentation. The equation consists of the AC equation with a fidelity term which enforces the solution to be the given data. In the proposed method, we start from a coarse grid and refine the grid until the accuracy of the data classification reaches a given tolerance. Therefore, we can avoid a laborious trial and error procedure. For a numerical method for the modified AC equation, we use a recently developed explicit hybrid scheme. We perform several 2D and 3D computational tests to demonstrate the performance of the proposed method. The computational results confirm that the proposed algorithm is automatic.

An efficient volume repairing method by using a modified Allen-Cahn equation

Classifying and rendering volumes of the structure are two essential goals of the visualization process. However, loss of some voxels can cause poor visualization results, such as small holes or non-smooth patches in visualized volumes. Beginning with the classified volumes, we propose a modified Allen-Cahn equation, which has the motion of mean curvature, to recover lost voxels and to fill holes. Consequently, a probability function can be obtained, which indicates the probability of each voxel being a volume voxel. Usually, the obtained probability function is smooth due to the motion of the mean curvature flow. Therefore visualization quality of volumes can be significantly improved. The equation is numerically computed by the unconditional stable operator splitting method with a large time step size. Thus the numerical scheme is fast and can be straightforwardly applied to GPU-accelerated DCT implementation that performs up to many times faster than CPU-only alternatives. Many experimental results have been performed to demonstrate the efficiency of the proposed method.

Shape transformation using the modified Allen–Cahn equation

In this paper, we propose a phase-field model for the shape transformation and its simple numerical method. The proposed phase-field model is based on the Allen–Cahn equation, which is a second-order nonlinear parabolic partial differential equation and originally arises from modeling phase separation in alloys. The numerical scheme is a hybrid method using the operator splitting method. To validate the proposed phase-field model for the shape transformation, we perform two- and three-dimensional shape metamorphosis. The proposed phase-field model is also compared with the linear interpolation. The computational results demonstrate that the proposed mathematical model satisfactorily and naturally simulate the shape transformation.

A phase field model for the propagation of electrical tree in nanocomposites

Electrical treeing is a main cause of long term degradation of polymeric insulation and inorganic nanofillers have been added to improve the treeing resistance of polymers. In this work, a phase field model for the propagation of electrical tree in nanocomposites is presented, in which the damage status of insulation is described with a spatially and time dependent continuous variable. The evolution of damage phase is described with a modified Allen-Cahn equation, which is driven by the free energies originated from the phase separation, interface and electric field. The electric field distribution during growth of electrical tree is obtained by solving the iterative form of Poisson's equation with the spectral iterative perturbation method. Then the phase field model is applied to investigate the influence of nanofillers shape and distribution on propagation process of electrical tree. The results indicate that the phase field model can reproduce the dendritic shape of electrical tree, and the parallel nanosheet presents superior performance in hindering electrical tree than nanoparticle and random nanosheet.

Existence of weak solution for mean curvature flow with transport term and forcing term

We study the mean curvature flow with given non-smooth transport term and forcing term, in suitable Sobolev spaces. We prove the global existence of the weak solutions for the mean curvature flow with the terms, by using the modified Allen-Cahn equation that holds useful properties such as the monotonicity formula.

Fast and Accurate Smoothing Method Using A Modified Allen–Cahn Equation

This paper presents a fast and accurate method using the Allen–Cahn (AC) equation with a fidelity term for curves smoothing of 2D shapes and volume smoothing of 3D shapes. The modified AC equation has a good smoothing dynamics and it is coupled with a fidelity term. The fidelity term forces the solution of the equation to be a close approximation to the original data. We use a hybrid explicit finite difference method to solve the equation. Therefore, we do not have any restriction on the shape of the computational domains. Several numerical tests for both the curve and surface smoothing problems are performed to demonstrate the robustness and efficiency of the proposed method. In particular, the proposed algorithm is useful for the 3D printing applications.

Image Segmentation Based on Modified Fractional Allen–Cahn Equation

We present the image segmentation model using the modified Allen–Cahn equation with a fractional Laplacian. The motion of the interface for the classical Allen–Cahn equation is known as the mean curvature flows, whereas its dynamics is changed to the macroscopic limit of Lévy process by replacing the Laplacian operator with the fractional one. To numerical implementation, we prove the unconditionally unique solvability and energy stability of the numerical scheme for the proposed model. The effect of a fractional Laplacian operator in our own and in the Allen–Cahn equation is checked by numerical simulations. Finally, we give some image segmentation results with different fractional order, including the standard Laplacian operator.

On a relation between the volume of fluid, level-set and phase field interface models

This paper discusses a relation between the re-initialization equation of the level-set functions derived by Wacławczyk [J. Comput. Phys., 299 (2015)] and the condition for the phase equilibrium provided by the stationary solution to the modified Allen-Cahn equation [Acta Metall., 27 (1979)]. As a consequence, the statistical model of the non-flat interface in the state of phase equilibrium is postulated. This new physical model of the non-flat interface is introduced based on the statistical picture of the sharp interface disturbed by the field of stochastic forces, it yields the relation between the sharp and diffusive interface models. Furthermore, the new techniques required for the accurate solution of the model equations are proposed. First it is shown, the constrained interpolation improves re-initialization of the level-set functions as it avoids oscillatory numerical errors typical for the second-order accurate interpolation schemes. Next, the new semi-analytical, second order accurate Lagrangian scheme is put forward to integrate the advection equation in time avoiding interface curvature oscillations introduced by the second-order accurate flux limiters. These techniques provide means to obtain complete, second-order convergence during advection and re-initialization of the interface in the state of phase equilibrium.

Triply periodic minimal surface using a modified Allen–Cahn equation

In this paper, we propose a fast and efficient model for triply periodic minimal surface. The proposed model is based on the Allen–Cahn equation with a Lagrange multiplier term. The Allen–Cahn equation has the motion of mean curvature. And the Lagrange multiplier term corresponding to the constant volume constraint also relates to the average of mean curvature. By combining two terms, the mean curvature will be constant everywhere on the surface at the equilibrium condition. The proposed numerical method with the second-order accuracy of time and space exhibits excellent stability. In addition, the resulting discrete system is solved by a fast numerical method such as a multigrid method. Various numerical experiments are performed to demonstrate the accuracy and robustness of the proposed method.

CONVERGENCE TO EQUILIBRIUM FOR A TIME SEMI-DISCRETE DAMPED WAVE EQUATION

We prove that the solution of the backward Euler scheme applied to a damped wave equation with analytic nonlinearity converges to a stationary solution as time goes to infinity. The proof is based on the Łojasiewciz-Simon inequality. It is much simpler than in the continuous case, thanks to the dissipativity of the scheme. The framework includes the modified Allen-Cahn equation and the sine-Gordon equation.