Narrow Band Domain Research Articles

An efficient and fast adaptive numerical method for a novel phase-field model of crystal growth

In this study, we present an efficient, fast, and fully explicit adaptive numerical scheme for solving a recently developed novel phase-field model of crystal growth in both two-dimensional (2D) and three-dimensional (3D) spaces. The novel phase-field model employed in this research incorporates a term specifically designed to eliminate the artificial curvature effect, thereby facilitating accelerated evolution when compared to traditional models. This enhancement significantly enhances the efficiency and speed of the simulation, leading to more expedited results. The crystal growth model employed in this study incorporates a phase-field equation to accurately represent the crystal interface, in addition to a heat equation that effectively models the distribution of temperature. To effectively solve the phase-field equation, we employ an adaptive numerical algorithm that optimizes the computational process. Our numerical scheme, specifically tailored for simulating dendritic growth, incorporates an adaptive narrow-band domain approach to accurately resolve the interfacial transition layer of the phase field. Furthermore, we enhance computational efficiency by implementing a double-sized grid for the temperature distribution, further improving the overall efficiency of the model. By combining these strategies, we achieve accurate and efficient solutions for the dendritic growth model. To validate the accuracy and efficiency of our proposed adaptive numerical method for solving the phase-field equation of dendritic growth, we conduct a series of numerical experiments in both 2D and 3D spaces. In these experiments, we assess the performance of our algorithm, analyzing its ability to accurately capture the intricate dynamics of dendritic growth while maintaining computational efficiency. By thoroughly evaluating the results obtained from these experiments, we provide strong evidence supporting the reliability and effectiveness of our adaptive numerical algorithm.

Numerical approximation of the square phase-field crystal dynamics on the three-dimensional objects

In the natural world, the phase field transition on the surface of a three-dimensional (3D) object is common. The square phase-field crystal (SPFC) equation is an effective model for describing the phase transition from liquid to solid with lattice structure. This study aims to present an efficient and practical computational scheme to simulate the SPFC dynamics on arbitrary surfaces. The surface is represented by the zero level-set of a singed distance function. The closest point method and appropriate pseudo-Neumann boundary condition are used to make the numerical solution to be constant along the normal direction to the arbitrary surface. In this sense, the calculations are transformed into a 3D discrete narrow computational band domain embedding the arbitrary surface and the finite difference method (FDM) is adopted to discretize the governing equation in 3D space. To maintain second-order spatial accuracy, the standard seven-point discrete Laplacian operator is applied to replace the discrete surface Laplace–Beltrami operator. We treat the nonlinear term in a totally explicit form and utilize the stabilization technique to suppress the stiffness of the numerical scheme. Therefore, we avoid solving any nonlinear equation or linear equation with variable coefficients. In each time step, we only need to update the linear system with constant coefficients. The proposed numerical scheme is highly efficient and its implementation is simple. Various numerical experiments demonstrate that the proposed numerical scheme not only achieves the expected accuracy but also works well for the simulations of phase transition and crystallization on surfaces with smooth and sharp shapes.

An Explicit Adaptive Finite Difference Method for the Cahn-Hilliard Equation.

In this study, we propose an explicit adaptive finite difference method (FDM) for the Cahn–Hilliard (CH) equation which describes the process of phase separation. The CH equation has been successfully utilized to model and simulate diverse field applications such as complex interfacial fluid flows and materials science. To numerically solve the CH equation fast and efficiently, we use the FDM and time-adaptive narrow-band domain. For the adaptive grid, we define a narrow-band domain including the interfacial transition layer of the phase field based on an undivided finite difference and solve the numerical scheme on the narrow-band domain. The proposed numerical scheme is based on an alternating direction explicit (ADE) method. To make the scheme conservative, we apply a mass correction algorithm after each temporal iteration step. To demonstrate the superior performance of the proposed adaptive FDM for the CH equation, we present two- and three-dimensional numerical experiments and compare them with those of other previous methods.

A simple and practical finite difference method for the phase-field crystal model with a strong nonlinear vacancy potential on 3D surfaces

• Efficient phase-field crystal model with a vacancy potential on 3D surfaces is developed. • Various phase transitions are well simulated on complex surfaces. • The closest point-type approach is adopted and its numerical implementation is efficient. The crystallization is a typical process in material science and this process can be described by the phase-field crystal type models. To investigate the dynamics of phase-field crystal model with a nonlinear vacancy potential on 3D surfaces, we herein propose a simple and practical finite difference method. The surfaces are defined by the zero level-set of signed distance functions and are included in a fully three-dimensional domain. By defining appropriate pseudo-Neumann boundary condition and using closest-point type method, the computation on surfaces are transformed into a three-dimensional narrow band domain containing the surfaces. Therefore, the finite difference method can be adopted to perform the spatial discretization. A linear semi-implicit discretization with stabilization technique in time is considered. In each time step, we only need to solve the elliptic type equations with constant coefficients. The numerical implementation is highly efficient. The numerical results indicate that the proposed method not only has desired accuracy but also works well for the pattern formations on various curved surfaces.

Efficient and practical phase-field method for the incompressible multi-component fluids on 3D surfaces with arbitrary shapes

The fluid dynamics on a curved surface can be used to approximately describe many physical phenomena in the natural world, such as the atmospherical circulation on a planet. The present work develops a simple, practical, and efficient numerical method for the multi-phase fluid simulations on arbitrarily curved surfaces in 3D space. The multi-component Cahn–Hilliard type phase-field fluid model is adopted to capture the motion of multiple interfaces. The velocity field and pressure are updated by solving the incompressible Navier–Stokes equations. To achieve the decoupled computations of pressure and velocity, the Chorin's projection method with pressure correction is considered. A linear semi-implicit time-marching scheme is designed for the phase-field equations. By adopting the embedding method and appropriate pseudo-Neumann boundary conditions, the direct calculations on the surface can be extended into a 3D narrow band domain including the surface. The standard Laplacian operator can be used to replace the surface Laplace–Beltrami operators and the finite difference method is used to perform the discretization in space. In each time step, all variables can be updated in a step-by-step manner because the proposed method is totally decoupled. Therefore, the whole algorithm is highly efficient and easy to implement. Extensive numerical experiments indicate that the proposed method has good performance for the multi-component simulations on curved surfaces with complex shapes.

Simple and practical method for the simulations of two-component PFC models for binary colloidal crystals on curved surfaces

Herein, we propose a practical numerical method for solving two-component phase-field crystal model with vacancy effect on curved surfaces and perform various numerical simulations. In the present work, the curved surfaces are defined by the zero level-set of signed distance functions. By adopting the pseudo-Neumann boundary condition and closest-point type method, we can extend the computation into a three-dimensional narrow band domain which contains the curved surfaces. The spatial discretization is performed by using the standard finite difference method and the seven-point Laplacian operator can be used to replace the surface Laplace–Beltrami operator. Therefore, the numerical implementation is simple and efficient. The semi-implicit discretization with stabilization term is considered to update the solutions in time. By simulating the pattern formations on curved surfaces with various shapes, we validate the good performance of the proposed method.

Numerical study of incompressible binary fluids on 3D curved surfaces based on the conservative Allen–Cahn–Navier–Stokes model

In this article, we propose a practical and highly efficient finite difference approach for two-phase fluid simulations on three-dimensional (3D) surfaces. The hydrodynamically coupled interfacial motion is captured by using the conservative Allen–Cahn–Navier–Stokes (CACNS) equations. By adopting the closest point method and the pseudo-Neumann boundary condition, the direct computations on curved surfaces are transferred to the 3D simulations in a narrow band domain embedding the surface. The projection method with pressure correction is used to decouple the computations of velocity and pressure. The operator splitting method is used to split the calculation of conservative Allen–Cahn equation into subproblems and the nonlinear part can be analytically solved. Therefore, the whole computation in each time iteration is highly efficient and easy to implement. The numerical experiments on various 3D curved surfaces are investigated to show the good performance of the proposed method.

A practical adaptive grid method for the Allen–Cahn equation

We present a simple and practical adaptive finite difference method for the Allen–Cahn (AC) equation in the two-dimensional space. We use a temporally adaptive narrow band domain embedded in the uniform discrete rectangular domain. The narrow band domain is defined as a neighboring region of the interface. We employ a recently developed explicit hybrid numerical scheme for the AC equation. Therefore, the computational algorithm on the narrow band discrete domain is simple and fast. We demonstrate the high performance of the proposed adaptive method for the AC equation through various computational experiments.

A fast and practical adaptive finite difference method for the conservative Allen–Cahn model in two-phase flow system

We present a simple and practical adaptive finite difference method for the conservative Allen–Cahn–Navier–Stokes system. For the conservative Allen–Cahn equation, we use a temporally adaptive narrow band domain embedded in the uniform discrete rectangular domain. The narrow band domain is defined as a neighboring region of the interface. The Navier–Stokes equation is solved in a fully discrete domain with the coarse grid than that for the CAC equation. Various benchmark numerical experiments, such as the pressure jump, droplet deformation in shear flow, falling droplet, and rising bubble, are performed to show that the proposed method is efficient and practical for the simulations of two-phase incompressible flow.

A phase-field model and its efficient numerical method for two-phase flows on arbitrarily curved surfaces in 3D space

Herein, we present a phase-field model and its efficient numerical method for incompressible single and binary fluid flows on arbitrarily curved surfaces in a three-dimensional (3D) space. An incompressible single fluid flow is governed by the Navier–Stokes (NS) equation and the binary fluid flow is governed by the two-phase Navier–Stokes–Cahn–Hilliard (NSCH) system. In the proposed method, we use a narrow band domain to embed the arbitrarily curved surface and extend the NSCH system and apply a pseudo-Neumann boundary condition that enforces constancy of the dependent variables along the normal direction of the points on the surface. Therefore, we can use the standard discrete Laplace operator instead of the discrete Laplace–Beltrami operator. Within the narrow band domain, the Chorin’s projection method is applied to solve the NS equation, and a convex splitting method is employed to solve the Cahn–Hilliard equation with an advection term. To keep the velocity field tangential to the surface, a velocity correction procedure is applied. An effective mass correction step is adopted to preserve the phase concentration. Computational results such as convergence test, Kevin–Helmholtz instability, and Rayleigh–Taylor instability on curved surfaces demonstrate the accuracy and efficiency of the proposed method.

A practical finite difference scheme for the Navier–Stokes equation on curved surfaces in [formula omitted

We present a practical finite difference scheme for the incompressible Navier–Stokes equation on curved surfaces in three-dimensional space. In the proposed method, the curved surface is embedded in a narrow band domain and the governing equation is extended to the narrow band domain. We use the standard seven-point stencil for the Laplace operator instead of a discrete Laplacian–Beltrami operator by using the closet point method and pseudo-Neumann boundary condition. The well-known projection method is used to solve the incompressible Navier–Stokes equation in the narrow band domain. To make the velocity field be parallel to the surface, a velocity correction step is used. Various numerical experiments, such as the divergence-free test, the convergence rate, and the energy dissipation, are performed on curved surfaces, which demonstrated that our proposed method is robust and practical.

A meshless technique based on generalized moving least squares combined with the second-order semi-implicit backward differential formula for numerically solving time-dependent phase field models on the spheres

In the current research paper, the generalized moving least squares technique is considered to approximate the spatial variables of two time-dependent phase field partial differential equations on the spheres in Cartesian coordinate. This is known as a direct approximation (it is the standard technique for generalized finite difference scheme [69,77]), and it can be applied for scattered points on each local sub-domain. The main advantage of this approach is to approximate the Laplace-Beltrami operator on the spheres using different types of distribution points simply, in which the studied mathematical models are involved. In fact, this scheme permits us to solve a given partial differential equation on the sphere directly without changing the original problem to a problem on a narrow band domain with pseudo–Neumann boundary conditions. A second-order semi-implicit backward differential formula (by adding a stabilized term to the chemical potential that is the second-order Douglas-Dupont-type regularization) is applied to approximate the temporal variable. We show that the time discretization considered here guarantees the mass conservation and energy stability. Besides, the convergence analysis of the proposed time discretization is given. The resulting fully discrete scheme of each partial differential equation is a linear system of algebraic equations per time step that is solved via an iterative method, namely biconjugate gradient stabilized algorithm. Some numerical experiments are presented to simulate the phase field Cahn-Hilliard, nonlocal Cahn-Hilliard (for diblock copolymers as microphase separation patterns) and crystal equations on the two-dimensional spheres.

A conservative finite difference scheme for the N-component Cahn–Hilliard system on curved surfaces in 3D

This paper presents a conservative finite difference scheme for solving the N-component Cahn–Hilliard (CH) system on curved surfaces in three-dimensional (3D) space. Inspired by the closest point method (Macdonald and Ruuth, SIAM J Sci Comput 31(6):4330–4350, 2019), we use the standard seven-point finite difference discretization for the Laplacian operator instead of the Laplacian–Beltrami operator. We only need to independently solve ($$N-1$$) CH equations in a narrow band domain around the surface because the solution for the Nth component can be obtained directly. The N-component CH system is discretized using an unconditionally stable nonlinear splitting numerical scheme, and it is solved by using a Jacobi-type iteration. Several numerical tests are performed to demonstrate the capability of the proposed numerical scheme. The proposed multicomponent model can be simply modified to simulate phase separation in a complex domain on 3D surfaces.

Numerical Simulation of Pattern Formation on Surfaces Using an Efficient Linear Second-Order Method

We present an efficient linear second-order method for a Swift–Hohenberg (SH) type of a partial differential equation having quadratic-cubic nonlinearity on surfaces to simulate pattern formation on surfaces numerically. The equation is symmetric under a change of sign of the density field if there is no quadratic nonlinearity. We introduce a narrow band neighborhood of a surface and extend the equation on the surface to the narrow band domain. By applying a pseudo-Neumann boundary condition through the closest point, the Laplace–Beltrami operator can be replaced by the standard Laplacian operator. The equation on the narrow band domain is split into one linear and two nonlinear subequations, where the nonlinear subequations are independent of spatial derivatives and thus are ordinary differential equations and have closed-form solutions. Therefore, we only solve the linear subequation on the narrow band domain using the Crank–Nicolson method. Numerical experiments on various surfaces are given verifying the accuracy and efficiency of the proposed method.

Fast and accurate adaptive finite difference method for dendritic growth

We propose a fast and accurate adaptive numerical method for solving a phase-field model for dendritic growth. The phase-field model for dendritic growth consists of two equations. One is for capturing the interface between solid and melt. The other is for the temperature distribution. For the phase-field equation, we apply a hybrid explicit method on a time-dependent narrow-band domain, which is defined using the phase-field function. For the temperature equation, we apply the explicit Euler method on the whole computational domain. The novelties of the proposed numerical algorithm are that it is very simple and that it does not require the conventional complex adaptive data structures. Our numerical simulation results are consistent with previous results. Furthermore, the computational time required (CPU time) is shorter.

Efficient 3D Volume Reconstruction from a Point Cloud Using a Phase-Field Method

We propose an explicit hybrid numerical method for the efficient 3D volume reconstruction from unorganized point clouds using a phase-field method. The proposed three-dimensional volume reconstruction algorithm is based on the 3D binary image segmentation method. First, we define a narrow band domain embedding the unorganized point cloud and an edge indicating function. Second, we define a good initial phase-field function which speeds up the computation significantly. Third, we use a recently developed explicit hybrid numerical method for solving the three-dimensional image segmentation model to obtain efficient volume reconstruction from point cloud data. In order to demonstrate the practical applicability of the proposed method, we perform various numerical experiments.

A tracking microwave wideband band reject filter using antiphase locking

ABSTRACTThis paper reports implementation of a wideband microwave band reject filter (BRF) at a centre frequency of 10 GHz which is based on a new principle of out of phase locking of a pair of Gunn oscillators by a reflected signal. The measured 3-dB bandwidth of the BRF is 486 MHz which is 4.86% of the centre frequency of the BRF with a rejection ratio of 7.08 dB. The bandwidth lies in between narrow band and ultrawide band domain.

Numerical simulation of the zebra pattern formation on a three-dimensional model

In this paper, we numerically investigate the zebra skin pattern formation on the surface of a zebra model in three-dimensional space. To model the pattern formation, we use the Lengyel–Epstein model which is a two component activator and inhibitor system. The concentration profiles of the Lengyel–Epstein model are obtained by solving the corresponding reaction–diffusion equation numerically using a finite difference method. We represent the zebra surface implicitly as the zero level set of a signed distance function and then solve the resulting system on a discrete narrow band domain containing the zebra skin. The values at boundary are dealt with an interpolation using the closet point method. We present the numerical method in detail and investigate the effect of the model parameters on the pattern formation on the surface of the zebra model.

A finite difference method for a conservative Allen–Cahn equation on non-flat surfaces

We present an efficient numerical scheme for the conservative Allen–Cahn (CAC) equation on various surfaces embedded in a narrow band domain in the three-dimensional space. We apply a quasi-Neumann boundary condition on the narrow band domain boundary using the closest point method. This boundary treatment allows us to use the standard Cartesian Laplacian operator instead of the Laplace–Beltrami operator. We apply a hybrid operator splitting method for solving the CAC equation. First, we use an explicit Euler method to solve the diffusion term. Second, we solve the nonlinear term by using a closed-form solution. Third, we apply a space–time-dependent Lagrange multiplier to conserve the total quantity. The overall scheme is explicit in time and does not need iterative steps; therefore, it is fast. A series of numerical experiments demonstrate the accuracy and efficiency of the proposed hybrid scheme.

Feature-driven topology optimization method with signed distance function

In this paper, a feature-driven topology optimization method is developed. This is the first study on layout design of multiple engineering features using level-set functions (LSFs) and Boolean operations. The novelty of this work is threefold. First, multiple engineering features of arbitrary shape are considered as basic design primitives and topology variation is achieved via the layout and shape optimization of the features. Kreisselmeier–Steinhauser (KS) function constructed by means of Boolean operations is adopted as the LSF, which uses an implicit function to ensure a smooth description and topological changes of basic features and the whole structure. Second, using a modified Heaviside function to smooth the void–solid material transition over a fixed computing mesh, a narrow-band domain integral scheme is developed for the efficient sensitivity analysis. Third, the gray material distribution regions at the feature-connecting portions are analyzed and the underlying reason for that is traced to the non-equidistant distribution of level-set contours of specific features. To avoid the gray regions, an approximated signed distance function is proposed to regularize the LSF and KS function. The bounded normalization property of the KS function is highlighted for its construction with the signed distance functions or normalized first-order approximations. Numerical examples are finally tested to demonstrate the validity and merits of the proposed feature-driven topology optimization for complicated design problems.