A KWC-type Multi-phase-field Model for Static Recrystallization through Subgrain Rotation and Coalescence

This repository provides the paper and source code for a KWC-type Multi-phase-field model designed to describe static recrystallization behavior associated with subgrain rotation and coalescence in polycrystalline materials.

Note: This paper summarizes the formulation and numerical examples associated with the KWC-type multi-phase-field model presented at the Asian Pacific Congress on Computational Mechanics (APCOM 2019, Taipei, Taiwan) under the title "Multi-phase-field Model Considering Evolution of Crystal Orientation for Nucleation from Subgrains in Polycrystal."

Paper

• mpfkwc_paper.pdf (preprint)

Overview

This study develops a KWC-type Multi-phase-field model in which multiple phase-field variables represent grains or subgrains, while crystal orientation is treated as an additional order parameter. The model is formulated based on a Ginzburg-Landau-type free energy functional and is intended to describe microstructural evolution involving grain-boundary migration, orientation evolution, and coalescence of neighboring subgrains.

The repository includes both two-dimensional and three-dimensional implementations:

• mpfkwc.py: two-dimensional implementation with scalar (angle) orientation field
• mpfkwc_3d.py: three-dimensional implementation using quaternion-based crystal orientation

Model Description

The KWC-type Multi-phase-field model introduces a phase-field order parameter $\phi_{[\iota]}$ for each grain $\iota$ ($\iota = 1, \ldots, N$) and a crystal-orientation order parameter $\theta$.

Free Energy Functional

The Ginzburg-Landau free-energy functional is defined as

$$F = \int_{\mathscr{V}} \left( f_{\mathrm{grad}} + f_{\mathrm{double}} + f_{\mathrm{drive}} + f_{\mathrm{orie}} \right) \, \mathrm{d}v,$$

where $f_{\mathrm{grad}}$ is the gradient energy of the order parameters, $f_{\mathrm{double}}$ is the double-well potential, $f_{\mathrm{drive}}$ is the driving-force energy associated with stored energy, and $f_{\mathrm{orie}}$ is the orientation-gradient energy. The orientation-gradient energy extends the KWC-type formulation to the multi-phase-field framework:

$$f_{\mathrm{orie}} = s\, G(\{\phi_{[\iota]}\}) \|\nabla\theta\| + \frac{1}{2}\varepsilon^2 H(\{\phi_{[\iota]}\}) \|\nabla\theta\|^2,$$

where $s$ and $\varepsilon$ are coefficients, and $G$ and $H$ are global weight functions that depend on the phase-field variables and representative grain orientations. These functions are chosen to take minimum values at grain boundaries, thereby localizing orientation changes in those regions.

Evolution Equation for $\phi_{[\iota]}$

The phase-field evolution equation is derived from the variational derivative of the free energy under the sum constraint, yielding

$$\frac{\partial \phi_{[\iota]}}{\partial t} = - \sum_{\kappa=1}^{N} \frac{M^{\phi}_{[\iota\kappa]}}{N} \Biggl\{ \sum_{\lambda=1}^{N} \left[ \frac{1}{2} \left( \alpha_{[\iota\lambda]}^2 - \alpha_{[\kappa\lambda]}^2 \right) \nabla^2\phi_{[\lambda]} + W_{[\iota\lambda]} \frac{\partial p(\phi_{[\iota]},\phi_{[\lambda]})}{\partial\phi_{[\iota]}} - W_{[\kappa\lambda]} \frac{\partial p(\phi_{[\kappa]},\phi_{[\lambda]})}{\partial\phi_{[\kappa]}} \right] + \frac{\delta f_{\mathrm{drive}}}{\delta\phi_{[\iota]}} - \frac{\delta f_{\mathrm{drive}}}{\delta\phi_{[\kappa]}} + s \left( \frac{\partial G}{\partial\phi_{[\iota]}} - \frac{\partial G}{\partial\phi_{[\kappa]}} \right) \|\nabla\theta\| + \frac{1}{2}\varepsilon^2 \left( \frac{\partial H}{\partial\phi_{[\iota]}} - \frac{\partial H}{\partial\phi_{[\kappa]}} \right) \|\nabla\theta\|^2 \Biggr\},$$

where $M^{\phi}_{[\iota\kappa]}$ is the phase-field mobility between grains $\iota$ and $\kappa$, $\alpha_{[\iota\lambda]}$ is the gradient energy coefficient, $W_{[\iota\lambda]}$ is the energy barrier coefficient, and $p(\phi_{[\iota]}, \phi_{[\lambda]})$ is the double-well potential. The last two terms represent contributions from the orientation-gradient energy $f_{\mathrm{orie}}$, which couples the orientation field to the phase-field evolution.　

Evolution Equation for $\theta$

The crystal-orientation field evolves according to

$$\frac{\partial \theta}{\partial t} = M^{\theta} \,\mathrm{div} \left[ \left( \frac{s\, G}{\|\nabla\theta\|} + \varepsilon^2 H \right) \nabla\theta \right],$$

where $M^{\theta}$ is the orientation mobility. The first-order term in $|\nabla\theta|$ localizes orientation changes near grain boundaries, while the second-order term smooths spatial variations in the orientation field.

In the three-dimensional extension, the scalar orientation $\theta$ is replaced by a unit quaternion field $q \in S^3$, and $|\nabla\theta|$ is replaced by $|\nabla q|$. A more detailed derivation is provided in the paper.

Program Correspondence

File	Dimension	Description
mpfkwc.py	2D	KWC-type Multi-phase-field model with scalar orientation field
mpfkwc_3d.py	3D	KWC-type Multi-phase-field model with quaternion-based orientation field

Installation

Clone this repository:

git clone https://github.com/your-name/mpf-kwc-model.git
cd mpf-kwc-model

Install the package and its dependencies using uv:

uv sync

Usage

Two-dimensional analysis

python mpfkwc.py

Three-dimensional analysis

python mpfkwc_3d.py

Example Results

Two-dimensional result

Distributions of grains and crystal orientations in static recrystallization analysis (Step No.: (i) 100, (ii) 600, (iii) 1000, (iv) 1700, (v) 2500). (a) Grain number and (b) crystal orientation.

Three-dimensional result

Distributions of grains and crystal orientations in static recrystallization analysis (Step No.: (i) 200, (ii) 600, (iii) 1200, (iv) 1800, (v) 2400). (a) Grain number and (b) IPF-colored crystal orientation.