This program solves the Cahn-Hilliard equation using an implicit pseudospectral method.

The Cahn-Hilliard equation is defined as

$$\frac{\partial c}{\partial t} = M \nabla^2\left[ \frac{\delta F}{\delta c}\right] = M \left[-\kappa \nabla^4 c + \nabla^2 f'(c)\right]$$

with M being a mobility and the functional of free-energy given by

$$F[c] = \int \left[ \frac{\kappa}{2} (\nabla c(\boldsymbol{r} ))^2 + f(c)\right] \text{d}{\boldsymbol{r}} $$

where $\kappa$ is a parameter related to the interfacial energy and $f$ is the bulk free-energy density given by

$$f(c) = W c^2(1-c)^2$$

where $W$ is the height of the thermodynamic barrier. The next Figure presents this bulk free-energy.

The concentration field can be expanded as a Fourier series in the form

$$\displaystyle c(\boldsymbol{r},t) = \frac{1}{L^2} \sum_{\boldsymbol{k}} \widehat{c}_{\boldsymbol{k}}(t) e^{i \boldsymbol{k} \cdot \boldsymbol{r} }$$

where the Fourier coefficients are given by

$$\widehat{c}_{\boldsymbol{k}}(t) = \mathcal{FT}[c(\boldsymbol{r},t) ] = \int_V c(\boldsymbol{r},t)e^{-i \boldsymbol{k} \cdot \boldsymbol{r} }\text{d}{\boldsymbol{r}} $$

and $k_i = {-\pi N_i/L_i, -\pi(N_i-1)/L_i, \ldots, \pi(N_i-1)/L_i,\pi N_i/L_i}$ where $\Delta_i$ is the gridsize of the meshgrid on the $i$ direction.

The Fourier transform of the dynamical equation is

and using an implicit Euler time integration, we have

such that

where $\Delta t$ is the time stepsize.

The following figures are results from the CH equations for a system with M=1.0, W=2.0, $\kappa=0.5$ and three different initial conditions $c_0 = 0.3, 0.5, 0.7$. The gridsize is $L = 64\pi$ with the number of gridpoints $N = 2^9 = 512$. The initial condition is given by a normal distribution

$$c(\boldsymbol{r},t=0) = c_0 + 0.1 \mathcal{N}(0,1),$$

and our system is evolved during 10000 steps with stepsize of dt=0.1.

In cahnhilliard.py we use just the Numpy package to do the fft. In cahnhilliard-pytorch.py we use the torch package to do the fft using the CUDA-capable NVIDIA GPU.

• NumPy is the fundamental package for scientific computing with Python.
• PyTorch is a high-level library for machine learning, with multidimensional tensors that can also be operated on a CUDA-capable NVIDIA GPU.
• Matplotlib is a comprehensive library for creating static, animated, and interactive visualizations in Python.

If you use cahnhilliard.py or cahnhilliard-pytorch.py in your work, please consider to cite it using the following reference:

Soares, E. do A., Barreto, A. G. & Tavares, F. W. Exponential Integrators for Phase-Field Equations using Pseudo-spectral Methods: A Python Implementation. 1–12 (2023). ArXiv: 2305.08998

Bibtex:

@article{Soares2023,
archivePrefix = {arXiv},
arxivId = {2305.08998},
author = {Soares, Elvis do A. and Barreto, Amaro G. and Tavares, Frederico W},
eprint = {2305.08998},
month = {may},
pages = {1--12},
title = {{Exponential Integrators for Phase-Field Equations using Pseudo-spectral Methods: A Python Implementation}},
url = {http://arxiv.org/abs/2305.08998},
year = {2023}
}

Elvis Soares: [email protected]

Universidade Federal do Rio de janeiro

School of Chemistry