### 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 [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.

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 language | English |
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Pages (from-to) | 11-25 |

Number of pages | 15 |

Journal | Computer-Aided Design |

Volume | 105 |

Early online date | 05 Jul 2018 |

DOIs | |

Publication status | Published - Dec 2018 |

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### Cite this

Fogg, H. J., Sun, L., Makem, J. E., Armstrong, C. G., & Robinson, T. T. (2018). Singularities in structured meshes and cross-fields.

*Computer-Aided Design*,*105*, 11-25. https://doi.org/10.1016/j.cad.2018.06.002