13.7 Coalescing Surface Elements

When using radiosity models for simulating the transport of particles above the wafer in the case where the length of the mean free path is greater than the size of the feature (cf. Section 12.3), two operations consume the most part of the computation time. The first operation is determining the visibility between all surface elements, which is an $O(n^2)$ operation, where $n$ denotes the number of surface elements extracted from the level set grid. The second operation is solving a certain system of linear equations, which leads to calculating the inverse of a matrix with $n^2$ elements, which is an $O(n^3)$ operation.

Obviously increasing the number of surface elements is not a remedy in cases where high resolution is required. High resolution is needed, e.g., near the trench opening, and the bottom of the trench, and for the simulation of micro-trenching and side wall push back. One approach is to devise a refinement and coarsening strategy for unstructured grids at the level of the level set implementation and the algorithms working on it. This, however, complicates the fast marching algorithm necessary for extending the speed function. Here a different approach was taken by coarsening the surfaces after having been extracted from the level set grid.

The algorithm works by walking down the list of surface elements extracted as the zero level set and calculating the angle $\alpha$ between two neighboring surface elements. Whenever $\vert\pi-\alpha\vert$ is below a certain threshold value of a few degrees, the neighboring elements are coalesced into one. After one sweep through the list, the algorithm can be reapplied for further coarsening. After $k$ coarsening sweeps, at most $2^k$ surface elements are coalesced into one. The resulting longer surface elements are used for the radiosity calculation, after which the fluxes are translated back from the coarsened elements to the original ones. Table 12.1 in Section 12.7 will show the relative speed up that was achieved in a typical deposition simulation.