paper.md

title	tags	authors	affiliations	date	bibliography
Cahn Hilliard Phase-Field Simulation by using CUDA GPU: Exascale Computational Materials Science	CUDA phase transformation numerical methods finite difference computational materials science	name orcid affiliation Mehrdad Yousefi 0000-0002-6497-7166 1	name index Department of Materials Science and Engineering, Clemson University, USA 1	1 November 2018	paper.bib

Summary

Cahn-Hilliard-CUDA is a computational materials science package, which aims to facilitate exascale phase transformation and materials design simulations. The main feature of Cahn-Hilliard-CUDA is porting phase-field simulation into the GPU (Graphical Processing Units) in order to speed up the simulations. The GPU computing program design needs clever memory allocation and transfer from/into CPU in order to achieve considerable speed up. As a result, all the numerical calculations in the Cahn-Hilliard-CUDA package are done within device kernels and only the at time of I/O the result will be transferred to the CPU. Also the Cahn-Hilliard-CUDA is written in a modular way which could facilitate future developments and include more complex materials design models.

Mathematics

The main theory behind the Cahn-Hilliard-CUDA is nonlinear diffusion equation based double-well chemical potential. This equation (i.e. Cahn-Hilliard equation) could be written as:

$$\frac{\partial c}{\partial t} = D \nabla^{2} (c^{3}-c-\gamma \nabla^{2} c)$$

Where $D$ is the diffusion coefficient and $\gamma$ controls interface thickness between two phases of different materials. In fact the nonlinear term of $\mu = c^{3}-c-\nabla^{2} c$ is the definition of chemical potential and is extracted from minimizing this Lyaponov free energy functional as:

$$F = \int_{v} [\frac{1}{4}(c^{2}-1)^{2} \frac{\gamma}{2} |\nabla c|^{2}] dv$$

Where $F$ is the free energy functional and $v$ is the volume of interest. In Cahn-Hilliard-CUDA the above equation is discretized by using Crank-Nicolson method in order to find the numerical solution as:

$$c_{new}^{t+1} = c_{old}^{t} + D \Delta t (\frac{\mu^{x+1,y,z}+\mu^{x-1,y,z}-2\mu^{x,y,z}}{\Delta x^{2}} + \frac{\mu^{x,y+1,z}+\mu^{x,y-1,z}-2\mu^{x,y,z}}{\Delta y^{2}} + \frac{\mu^{x,y,z+1}+\mu^{x,y,z-1}-2\mu^{x,y,z}}{\Delta z^{2}})$$

Figures

Spinodal decomposition of two immiscible materials:

Acknowledgements

We acknowledge Palmetto cluster of Clemson University, USA for giving us the time to run the simulations.

References