We may also transition between the hex and tet mesh using square-base pyramids. Geode-template quality is acceptable in some limited regimes. The STC, described in Pete Murdoch's PhD thesis, is the observation that the dual of a hexahedral mesh can be viewed as an arrangement of surfaces, each surface representing the layer of hexes it passes through. In addition to Whisker Weaving, the STC has been used to create the geode-template, some mesh connectivity improvement algorithms, and aids the understanding of mesh refinement. We have developed both a state-list algorithm that advances the front, building the STC arrangment, one hex at a time, and a curve contraction algorithm that builds the STC arrangment one surface at a time.
Local rules guide the advancing front away from poor quality degeneracies.
After the arrangment is build, the STC is moved and refined in an effort to improve quality. Although a topologically well-defined mesh is always created, acceptable quality meshes remain rare. Even so, no good-quality hexahedral mesh has been found by the research community, by any means, for some surprisingly small and simple examples.