SNOW: Robust simulation of strongly anisotropic Cahn-Hilliard systems

Purpose

• The package "Snow" solves the strongly anisotropic Cahn-Hilliard system (regularized by the nonlinear Willmore functional) on a 2-D square domain. It uses a Legendre-Galerkin method for systems of coupled elliptic equations in space and a first-order energy stabilized finite difference scheme in time.

Specifications

• Name: Snow.
• Author: Feng Chen.
• Finishing date: 11/10/2011.
• Languages: Fortran 90, MATLAB.
• Required libraries: BLAS, LAPACK.

Simple Example

• Equation: $$\begin{equation} \left\{ \begin{aligned} & \phi_{t} =\frac{1}{\epsilon}\Delta\mu, && \quad \text{in } \Omega, \\ & \mu =\frac{\delta\mathcal E}{\delta\phi}, && \quad \text{in } \Omega, \\ & \omega =\frac{1}{\epsilon}f'(\phi)-\epsilon\Delta\phi, && \quad \text{in } \Omega, \\ & \frac{\partial \phi}{\partial \boldsymbol n} = \frac{\partial \mu}{\partial \boldsymbol n} = \frac{\partial \omega}{\partial \boldsymbol n} = 0, && \quad \text{on } \partial\Omega , \end{aligned} \right. \end{equation}$$ where $$\begin{equation} \begin{aligned} & \mathcal E(\phi)= \int_{\Omega}(\mathcal F+ \frac{\beta}{2} \mathcal G)\text{d} \Omega,\\ & \mathcal F(\phi)= \frac{\gamma(\boldsymbol n)}{\epsilon}(f(\phi)+\frac{\epsilon^{2}}{2}|\nabla\phi|^{2}) ,\\ &\mathcal G (\phi) = \omega^2,\\ &f(\phi) = \frac{1}{4} (\phi^2-1)^2,\\ &\omega=\frac{1}{\epsilon}f'(\phi)-\epsilon\Delta\phi, \\ & \gamma ( \boldsymbol n) =1+\alpha\cos(4\theta)= 1 + \alpha (4 \sum_{i=1}^d n_i^4 -3). \end{aligned} \end{equation}$$
• Initial conditions: $$\begin{equation} \begin{aligned} &\phi_0 = \tanh (\frac{\sqrt{x^2+y^2}-r}{\epsilon}), \\ &\omega_0 = f'(\phi_0). \end{aligned} \end{equation}$$
• Parameters: $$\begin{equation} \Omega = (-1,1)^2 , \, T=0.15, \, r=0.5, \, \epsilon=0.02, \, \beta=5\times10^{-4}, \, \alpha=0.3. \end{equation}$$

Quick Start

• Compiling and running:

  cd ./Lion
  make library
  ifort tst_Snow.f90 -llibrary -llapack -lblas
  ./a.out
  matlab plot_Snow.m
  

• Graphics: The left panel shows the evolution of the anisotropic dynamics in $\Omega$, and the right one shows the decreasing energy against time steps.

References

• Feng Chen and Jie Shen. Efficient energy stable schemes with spectral discretization in space for anisotropic Cahn-Hilliard systems, Communications in Computational Physics, Volume 13, Number 5, 1189-1208, (2013).
• Feng Chen and Jie Shen. Efficient spectral-Galerkin methods for systems of coupled second-order equations and their applications, Journal of Computational Physics, Volume 231, Issue 15, 5016-5028, (2012).

Code Highlights

!! Assign initial conditions
do i = 0, nx
 do j = 0, ny
  ! quadrature points
  x = (zx(i)*(x2-x1) + (x2+x1))/2d0
  y = (zy(j)*(y2-y1) + (y2+y1))/2d0
  select case (SnowAnisoCase) 
   case (1)  ! one circle
    phi(i,j) = tanh((sqrt(x*x+y*y)-r)/SnowEps)
   case (2)  ! two circles
    phi(i,j) = tanh((sqrt((x+1d0/2d0)**2+                   &
                          (y-1d0/2d0)**2)-1d0/4d0)/SnowEps) &
             + tanh((sqrt((x-1d0/5d0)**2+                   &
                          (y+1d0/5d0)**2)-1d0/2d0)/SnowEps)
    phi(i,j) = phi(i,j) - 1d0
  end select 
 end do
end do     
omega =- (phi*phi*phi-phi)/SnowEps