Singularities in structured meshes and cross-fields. (December 2018) - Fogg, Harold J.; Sun, Liang; Makem, Jonathan E.; Armstrong, Cecil G.; Robinson, Trevor T.
Abstract: Singularities in structured meshes are vertices that have an irregular valency.The integer irregularity in valency is called the singularity index of the vertex of the mesh. Singularities in cross-fields are closely related which are isolated points where the cross-field vectors are defined in its limit neighbourhood but not at the point itself. For a closed surface the genus determines the minimum number of singularities that are required in a structured mesh or in a cross-field on the surface. Adding boundaries and forcing conformity of the mesh or alignment of the cross-field to them also affects the minimum number of singularities required. In this paper a simple formula is derived from Bunin's Continuum Theory for Unstructured Mesh Generation (Bunin, 2008) that specifies the net sum of singularity indices that must occur in a cross-field with even numbers of vectors on a face or surface region with alignment conditions. The formula also applies to mesh singularities in quadrilateral and triangle meshes and the correspondence to 3-D hexahedral meshes is related. Some potential applications are discussed. Highlights: A formula is presented for the net sum of singularity indices in cross-fields and meshes. The formula is only related to the vertex types and the face's Euler Characteristic. A new condition on the validity of line singularity networks in hex meshes is described. Quad mesh singularities can be located by an offsetting approach.