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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

\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}

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

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

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.
\includegraphics [height=0.62\textheight]{ppl/locos.eps}

Peter Fleischmann