4.1.1 Algebraic Method

The parametrized boundary surfaces are algebraically combined to create the interior grid. The mapping of the unit square/cube with coordinates $u,v,w$ onto the domain is performed by the transfinite interpolation [184]. In two dimensions it can be written as

$$\begin{eqnarray*} \mathbf{M}(u,v) & = & (1-v)\mathbf{B}_{bottom}(u) + u\mathbf{B... ...u(1-v)\mathbf{V}_{2} - uv\mathbf{V}_{3} - (1-u)v\mathbf{V}_{4} \end{eqnarray*}$$

where $\mathbf{B}$ are the boundaries, $\mathbf{V}$ the four corner vertices, and $\mathbf{M}$ is the mapping function to calculate the coordinates $x,y$ in the real domain. The mapping function satisfies

$$\begin{eqnarray*} \mathbf{M}(u,0) & = & \mathbf{B}_{bottom}(u) \\ \mathbf{M}(u,... ...) & = & \mathbf{V}_{3} \\ \mathbf{M}(0,1) & = & \mathbf{V}_{4} \end{eqnarray*}$$

The resulting grid is useful if the boundaries are convex or not too twisted. A three-dimensional example of the mesh of a LOCOS which was generated with a package described in [13] is shown in Fig. 4.2-a.

Figure 4.2: LOCOS: (a) structured mesh, 2000 tetrahedra (b) unstructured mesh, 957 tetrahedra.