Title: Solving phase field equations using a meshless method

Authors: J X Zhou and M E Li

Source: Communications in numerical methods in engineering, Vol. 22, Issue 11, pp. 1109-1115.


The phase field equation is solved by using a meshless reproducing kernel particle method (RKPM) for the very first time. The 1D phase field equation is solved using different grid sizes and various time steps at a given grid size. The method can give accurate solutions across the interface, and allows a larger time step than explicit finite-difference method. The 2D phase field equation is computed by the present method and a classic shrinking of a circle is simulated. This shows the powerfulness and the potential of the method to treat more complicated problems.

Title: Density functional studies of multiferroic magnetoelectrics

: Nicola A Hill

Source: Annual Review of Materials Research, Vol. 32: 1-37 (Volume publication date August 2002)


Multiferroic magnetoelectrics are materials that are both ferromagnetic and ferroelectric in the same phase. As a result, they have a spontaneous magnetization that can be switched by an applied magnetic field and a spontaneous polarization that can be switched by an applied electric field. In this paper we show that density functional theory has been invaluable both in explaining the properties of known magnetically ordered ferroelectric materials, and in predicting the occurrence of new ones. Density functional calculations have shown that, in general, the transition metal d electrons essential for magnetism reduce the tendency for off-center ferroelectric distortion. Consequently, an additional electronic or structural driving force must be present for ferromagnetism and ferroelectricity to occur simultaneously.

Everybody needs an iterative solver at some point or other. Recently, I sent a few links to a friend of mine. I thought I will also maintain the list in this page for future use.

In case you know of any good resources that I have missed here, leave a note!

Title: Stability analysis of the Crank-Nicholson method for variable coefficient diffusion equation

Author: Charles Tadjeran

Source: Communications in Numerical Methods in Engineering, Vol 23, Issue 1, pp. 29-34, 2006.

The Crank-Nicholson method is a widely used method to obtain numerical approximations to the diffusion equation due to its accuracy and unconditional stability.

When the diffusion coefficient is not a constant, the general approach is to obtain a discretization for the PDE in the same manner as the case for constant coefficients. In this paper, we show that the manner of this discretization may impact the stability of the resulting method and could lead to instability of the numerical solution. It is shown that the classical Crank-Nicholson method will fail to be unconditionally stable if the diffusion coefficient is computed at the time gridpoints instead of at the midpoints of the temporal subinterval. A numerical example is presented and compared with the exact analytical solution to examine its divergence.

Title: A discretization of the n-dimensional Laplacian for a dimension-independent stability limit

Author(s): Anand Kumar

Source: Proceedings: Mathematical, Physical and Engineering Sciences, Vol. 457, No. 2015, (Nov. 8, 2001), pp. 2667-2674.

A discretization of the n-dimensional Laplacian employing all the 3^n points of the finite-difference stencil has been presented. Using this discretization, the stability limit of the heat conduction and the wave equations are found to become dimension-independent \frac{1} {2} and 1, respectively.

Title: Spectral implementation of an adaptive moving mesh method for phase-field equations

Authors: W M Feng, P Yu, S Y Hu, Z K Liu, Q Du and L-Q Chen

Source: Journal of Computational Physics, Vo. 220, Issue 1, 20 December 2006, pp. 498-510.


Phase-field simulations have been extensively applied to modeling microstructure evolution during various materials processes. However, large-scale simulations of three-dimensional (3D) microstructures are still computationally expensive. Among recent efforts to develop advanced numerical algorithms, the semi-implicit Fourier spectral method is found to be particularly efficient for systems involving long-range interactions as it is able to utilize the fast Fourier transforms (FFT) on uniform grids. In this paper, we report our recent progress in making grid points spatially adaptive in the physical domain via a moving mesh strategy, while maintaining a uniform grid in the computational domain for the spectral implementation. This approach not only provides more accurate treatment at the interfaces requiring higher resolution, but also retains the numerical efficiency of the semi-implicit Fourier spectral method. Numerical examples using the new adaptive moving mesh semi-implicit Fourier spectral method are presented for both two and three space dimensional microstructure simulations, and they are compared with those obtained by other methods. By maintaining a similar accuracy, the proposed method is shown to be far more efficient than the existing methods for microstructures with small ratios of interfacial widths to the domain size.

Title: Numerical study of the Cahn-Hilliard equation in one, two and three dimensions

Authors: E V L de Mello and Otton Teixeira da Silveira Filho

Source: Physica A: Statistical and theoretical physics, Vol. 347, 1 March 2005, pp. 429-443.


The Cahn–Hilliard (CH) equation is related with a number of interesting physical phenomena like the spinodal decomposition, phase separation and phase ordering dynamics. On the other hand this equation is very stiff and the difficulty to solve it numerically increases with the dimensionality and therefore, there are several published numerical studies in one dimension (1D), dealing with different approaches, and much fewer in two dimensions (2D). In three dimensions (3D) there are very few publications, usually concentrate in some specific result without the details of the used numerical scheme. We present here a stable and fast conservative finite difference scheme to solve the CH with two improvements: a splitting potential into an implicit and explicit in time part and the use of free boundary conditions. We show that gradient stability is achieved in one, two and three dimensions with large time marching steps than normal methods.

Title: Conservative nonlinear difference scheme for the Cahn-Hilliard equation (Parts I and II)

Authors: S M Choo and S K Chung (Part I); S M Choo, S K Chung and K I Kim(Part II)

Source: Part I — Computers and Mathematics with applications, Vol. 36, Issue 7, October 1998, pp. 31-39; Part II — Computers and Mathematics with applications, Vol . 39, Issues 1-2, January 2000, pp. 229-243.


Part I:

Numerical solutions for the Cahn-Hilliard equation is considered using the Crank-Nicolson type finite difference method. Existence of the solution for the difference scheme has been shown by Brouwer fixed-point theorem. Stability, convergence and error analysis of the scheme are shown. We also show that the scheme preserves the discrete mass, even though the linearized scheme in [1] is conditionally stable and does not preserve the mass.

Part II:

A nonlinear conservative difference scheme is considered for the two-dimensional Cahn-Hilliard equation. Existence of the solution for the finite difference scheme has been shown and the corresponding stability, convergence, and error estimates are discussed. We also show that the scheme preserves the discrete total mass computationally as well as analytically.

Phase field models are used, among other things, for a study of grain growth [1,2]. In one type of phase field models used to study grain growth, each grain orientation is represented by an order parameter \eta_{i}, where the index i runs from 1 to N, where N is the total number of orientations present in the system [3,4]. Though this method is simple to implement and has been used extensively, it is computationally intensive. So, recently there had been at least two attempts to come up with an efficient numerical implementation:

  1. Efficient numerical algorithm for multiphase field simulations. Srikanth Vedantam and B S V Patnaik, Phys Rev E 73, 016703, 2006.

    Phase-field models have emerged as a successful class of models in a wide variety of applications in computational materials science. Multiphase field theories, as a subclass of phase-field theories, have been especially useful for studying nucleation and growth in polycrystalline materials. In theory, an infinite number of phase-field variables are required to represent grain orientations in a rotationally invariant free energy. However, limitations on available computational time and memory have restricted the number of phase-field variables used in the simulations. We present an approach by which the time and memory requirements are drastically reduced relative to standard algorithms. The proposed algorithm allows us the use of an unlimited number of phase-field variables to perform simulations without the associated burden on computational time or memory. We present the algorithm in the context of coalescence free grain growth.

  2. Sparse data structure and algorithm for the phase field method. J Gruber, N Ma, Y Wang, A D Rollett, and G S Rohrer, Modelling and Simulation in Materials Science and Engineering, 14, 1189, 2006.

    The concepts of sparse data structures and related algorithms for phase field simulations are discussed. Simulations of polycrystalline grain growth with a conventional phase field method and with sparse data structures are compared. It is shown that memory usage and simulation time scale with the number of nodes but are independent of the number of order parameters when a sparse data structure is used.

The source code for a C++ implementation of the method described in the paper of Gruber et al is available for download here (for non-profit scientific research purposes).

The idea behind these implementations is rather simple. Consider an arbitrary mesh point in a simulation cell. The mesh point either lies in the bulk of a given grain, or it lies in the interface. If it lies in the bulk, all the order parameters except the one corresponding to the grain orientation are zero, and there is nothing much to be done about the calculation at that point. On the other hand, if it lies in the interface, the total number of order parameters which have non-zero values are still a very small number as compared to the total number of orientations present in the entire system. Hence, if there is a database such that for any point (and its neighbours) we have the information of non-zero order parameters, then the calculation can be made more efficient.

Take a look and have fun!


[1] Phase field methods for microstructure evolution. Long-Qing Chen, Annu. Rev. Mater. Res., 32, 113, 2002.

[2] Modeling grain boundaries using a phase-field technique. J A Warren, R Kobayashi, and W Craig Carter, J Cryst Growth, 211, 1, 18, 2000.

[3] A novel computer simulation technique for modeling grain growth. Long-Qing Chen, Scripta Metallurgica et Materialia, 32, 1, 115, 1995.

[4] A phase field concept for multiphase systems. I Steinbach, F Pezzolla, B Nestler, M Seeszelberg, R Prieler, G J Schmitz, and J L L Rezende, Physica D, 94, 3, 135, 1996.

Here is a follow-up paper on the unconditionally stable time step for diffuse interface methods.

Paper: Maximally fast coarsening algorithms

Authors: Mowei Cheng and Andrew D. Rutenberg


We present maximally fast numerical algorithms for conserved coarsening systems thatare stable and accurate with a growing natural time step \Delta t = A t_{s}^{2/3}. We compare the scaling structure obtained from our maximally fast conserved systems directly against the standard fixed time-step Euler algorithm, and find that the error scales as \sqrt{A}—so arbitrary accuracy can be achieved. For non-conserved systems, only effectively finite time steps are accessible for similar unconditionally stable algorithms.