Geometrically adaptive numerical integration

Numerical integration over solid domains often requires geometric adaptation to the solid's boundary. Traditional approaches employ hierarchical adaptive space decomposition, where the integration cells intersecting the boundary are either included or discarded based on their position with respect to the boundary and/or statistical measures. These techniques are inadequate when accurate integration near the boundary is particularly important. In boundary value problems, for instance, a small error in the boundary cells can lead to a large error in the computed field distribution. We propose a novel technique for exploiting the exact local geometry in boundary cells. A classification system similar to marching cubes is combined with a suitable parameterization of the boundary cell's geometry. We can then allocate integration points in boundary cells using the exact geometry instead of relying on statistical techniques. We show that the proposed geometrically adaptive integration technique yields greater accuracy with fewer integration points than previous techniques.