Driving force extension method - hands-on tutorial

This short tutorial is used at the ChiMaD Phase Field Workshop XVI, March 19-21, 2024. The goal is to provide an introduction to the driving force extension method in our recent paper:

Jin Zhang, Alexander F. Chadwick, David L. Chopp, Peter W. Voorhees, "Phase Field Modeling with Large Driving Forces," npj Computational Materials, 9 (2023) 166. doi: 10.1038/s41524-023-01118-0

Here, a 1D KKS model is used for demonstration. Several assumptions are made for simplification so that we can focus on the problem of the large driving force and the driving force extension method.

Driving force extension method

The driving force extension method is used to solve the stability problem resulting from a large driving force in phase field modeling.

Phase field equation typically has the following form

$$\tau\frac{\partial \phi}{\partial t} = \kappa \nabla^2{\phi} - m g'(\phi) - p'(\phi) F,$$

where $F$ is the driving force. If the driving force is much larger than the surface energy $F \gg \sigma/l$ (here, $\sigma$ is surface energy, $l$ is diffusion interface width), phase field simulation can suffer numerical instability. To solve this problem, a smaller interface width $l$ (and hence a smaller grid size) is needed. However, this limits the system size that one can simulate.

To solve the problem of the large driving force but still use a large grid size, the driving force extension method introduces a simple modification of the phase field equation

$$\tau\frac{\partial \phi}{\partial t} = \kappa \nabla^2{\phi} - m g'(\phi) - p'(\phi) \mathcal{P}(F),$$

where $\mathcal{P}$ projects $F$ to a constant perpendicular to the interface:

$$\mathcal{P}(F(\vec{x}, t)) = F(\vec{x}_\Gamma, t).$$

Here, $\vec{x}_\Gamma$ is the closest point on the interface $\Gamma$:

$$\vec{x}_\Gamma(\vec{x}) = \{\vec{y} : \min_{\vec{y}\in \Gamma} |{\vec{y}-\vec{x}}|\}$$

The algorithm for the projection $\mathcal{P}(F)$ is well-developed in the level-set community and is called velocity-extension.

Using the code

You will need Python 3 installed.

If you don't have Python, you can use Colab (Google account is needed)

Also, you need numpy and matplotlib. Install them by

pip install numpy matplotlib

To use the jupyter notebook, you need to install jupyter by

pip install jupyterlab

To open the jupyter notebook, run in the terminal

jupyter-lab

A browser window should automatically open.

Files

• driving_force_extension.ipynb : tutorial notebook

• driving_force_extension.py : python script in case you don't have jupyter

The python script is generated by

jupyter nbconvert --to script driving_force_extension.ipynb

Functions in the code

• dg : derivative of the double well (times 0.5)
• laplacian_1d : Laplacian (cell-based)
• apply_bc : apply boundary condition
• velext_simple : a simple velocity extension in 1D
• velext_fim : a general velocity extension in 1D
• kks1d : the KKS model

Citing our work

If you use the driving force extension method in your work, please cite our paper

Zhang, J., Chadwick, A.F., Chopp, D.L. et al. Phase field modeling with large driving forces. npj Comput Mater 9, 166 (2023).

You can use the BibTeX

@article{Zhang2023,
  title={Phase field modeling with large driving forces},
  author={Zhang, Jin and Chadwick, Alexander F and Chopp, David L and Voorhees, Peter W},
  journal={npj Computational Materials},
  volume={9},
  number={1},
  pages={166},
  year={2023},
  publisher={Nature Publishing Group UK London}
}

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For questions, comments, suggestions, or bug reports, email me at jzhang@northwestern.edu or visit github.