4 Setting of the Initial Order Parameter Profile and Initial Grid Refinement $\Lambda _{h}(0)$

The initial order parameter profile depends on the initial shape of the void $\Gamma(0)$ and can be expressed as

$$\phi(x,y)= \begin{cases}+1& \text{if} \quad d > \frac{\epsilon\pi... ...{\epsilon\pi}{2}, \ -1& \text{if} \quad d < -\frac{\epsilon\pi}{2} \end{cases}$$ (263)

Where $d=dist(P(x,y),\Gamma(0))$ is the signed normal distance of the point from the initial interface $\Gamma(0)$. To obtain sufficient resolution of this initial profile, the basic grid $T_{h}$ is transformed into grid $\Lambda _{h}(0)$ obeying the following initial grid refinement criterion (IGRC) for the circular void with center $O$ and radius $r$:

$$\vert dist(C_{K},O)-r\vert\leq \frac{\epsilon\pi}{2} \quad \wedge \quad h_{K} > \frac{\epsilon\pi}{n}$$ (264)

$n$ is the chosen number of grid elements across the void-metal interface width, $h_{K}$ is the longest vertex of the triangle $K$, and $C_{K}$ is its center of gravity. Now an adaptive algorithm defined in Section 4.5.7 transforms the basic grid $T_{h}$ into an initial grid $\Lambda _{h}(0)$ according to $IGRC$.