Singularities in structured meshes and cross-fields

Harold J. Fogg, Liang Sun, Jonathan E. Makem, Cecil G. Armstrong, Trevor T Robinson

Research output: Contribution to journalArticlepeer-review

14 Citations (Scopus)
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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 [1] 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.
Original languageEnglish
Pages (from-to)11-25
Number of pages15
JournalComputer-Aided Design
Early online date05 Jul 2018
Publication statusPublished - Dec 2018


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