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 |
|---|---|
| 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