6 Maintaining the Grid during Simulation

After an order parameter was evaluated on the $\Lambda_{h}(t_{n})$ a grid needs to be adapted according to the new void-metal interface position. Therefore it is necessary to extract all elements which are cut by the void-metal interface in grid $\Lambda_{h}(t_{n})$. The following condition is used: We take a triangle $K \in \Lambda_{h}(t_{n})$ and denote its vertices as $P_{0},P_{1},P_{2}$. The triangle $K$ belongs to the interfacial elements if for the values of the order parameter $\phi$ at the triangle's vertices holds $\phi(P_{0})\phi(P_{1})<0$ or $\phi(P_{1})\phi(P_{2})<0$. We assume that an interface intersects each edge of the element only once. The set of all interfacial elements at the time $t_{n}$ is denoted as $E(t_{n})$. The centers of gravity of each triangle from the $E(t_{n})$ build the interface point list $L(t_{n})$. The distance of the arbitrary point $Q$ from $L(t_{n})$ is defined as,

$$dist(Q,L(t_{n}))= \underset{P \in L(t_{n})}{min} dist(Q,P).$$ (270)

Thus we can define the transitional grid refinement criterion TGRC

$$dist(C_{K},L(t_{n})) \leq \frac{\epsilon\pi}{2}\quad \wedge \quad h_{K} > \frac{\epsilon\pi}{n}$$ (271)

The grid adaption for the next time step evaluation of the order parameter $\phi$ is now completed with respect to $TGRC$ as defined in the next section.

Figure 4.6: Atomic refinement operation.