Example 2: homogeneous nucleation

Files

• Comprehensive test file: main.cpp
• Reference results for comparison (r=0.99, t=0.1): time_specialized.csv
• Reference results for comparison (r=1, t=0.1): time_specialized.csv
• Reference results for comparison (r=1.01, t=0.1): time_specialized.csv
• Reference results for comparison (r=0.99, t=100): time_specialized.csv
• Reference results for comparison (r=1, t=100): time_specialized.csv
• Reference results for comparison (r=1.01, t=100): time_specialized.csv

Statement of the problem

This test corresponds to the homogeneous nucleation test proposed on PFhub, for single circular seed.

The domain $\Omega$ is a square $[0,100]\times[0,100]$

$$\begin{align} \frac{\partial \phi}{\partial t}&=-M (F'(\phi) - \lambda \Delta \phi - \Delta f p'(\phi)) \text{ in }\Omega \end{align}$$

where $\phi$ is the phase indicator, $F'$ the derivative against $\phi$ of the potential $F$ defined by:

$$\begin{align} F(\phi)&=\omega\phi^2(1-\phi)^2 \end{align}$$

$\Delta f$ is the nucleation driving force defined as a function of the critical radius $r^\star$:

$$\begin{align} \Delta f&=\dfrac{\sqrt{2}}{6 r^\star} \end{align}$$

Initial condition

The initial condition is given by:

$$\phi = \dfrac{1}{2} - \dfrac{1}{2} \tanh\left(\dfrac{r-r_0}{\sqrt{2}}\right).$$

Parameters used for the test

For this test, all phase-field parameters are equal to one.

Description	Symbol	Value
mobility coefficient	$M_\phi$	$1.0$
energy gradient coefficient	$\lambda$	$1$
depth of the double-well potential	$\omega$	$1$
radius of the initial circular seed	$r_0$	[$0.99r^\star$,$r^\star$,$1.01r^\star$]
critical radius	$r^\star$	$5$

Boundary conditions

Periodic boundary conditions are prescribed on boundary of the domain.

Numerical scheme

• Time integration: Euler Implicit over the interval $t\in[0,100]$ with a time-step $\delta t=10^{-2}$.
• Spatial discretization for convergence analysis: uniform grid with $N={256}$ nodes in each spatial direction, with $\mathcal{Q}_1$ finite elements
• Newton solver: relative tolerance $10^{-10}$, absolute tolerance $10^{-14}$
• Iterative solver: HYPRE_GMRES
• Preconditioner: HYPRE_ILU

Results

Figures 1 shows the evolution of the normalized fraction and energy density. The results are in good agreement with those presented in ¹ (see Figures 3 and 4 in the reference).

Figure 1: normalized fraction and energy density

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1. W. Wu, D. Montiel, J.E. Guyer, P.W. Voorhees, J.A. Warren, D. Wheeler, L. Gránásy, T. Pusztai, and O.G. Heinonen. Phase field benchmark problems for nucleation. Computational Materials Science, 193:110371, 2021. URL: https://www.sciencedirect.com/science/article/pii/S0927025621000963, doi:https://doi.org/10.1016/j.commatsci.2021.110371. ↩