Refinable C1 spline elements for irregular quad layout

Abstract

Building on a result of U. Reif on removable singularities, we construct C1 bi-3 splines that may include irregular points where less or more than four tensor-product patches meet. The resulting space complements PHT splines, is refinable and the refined spaces are nested, preserving for example surfaces constructed from the splines. As in the regular case, each quadrilateral has four degrees of freedom, each associated with one spline and the splines are linearly independent. Examples of use for surface construction and isogeometric analysis are provided.

Improved shape for refinable surfaces with singularly parameterized irregularities

“…Since the aim is to improve the shape of constructions such as [8], we consider the same type of input mesh whose nodes are interpreted as B-spline-like control points. That is, we consider as input a network of quadrilateral facets or quads .…”

“…Geometrically smooth ( G k ) constructions fit directly into the CAD pipeline and have been shown to deliver class A shape; but refinement requires careful book-keeping to record what edges are G -connected in the initial surface. This paper explores a third option: singularly-parameterized freeform surfaces as presented in [2, 3, 4, 5, 6, 7, 8]. [8] demonstrated simple uniform refinability of singularly-parameterized surfaces for iso-geometric computations on the surfaces at different levels of resolution.…”

“…This paper explores a third option: singularly-parameterized freeform surfaces as presented in [2, 3, 4, 5, 6, 7, 8]. [8] demonstrated simple uniform refinability of singularly-parameterized surfaces for iso-geometric computations on the surfaces at different levels of resolution. However, its surfaces, as well as all the earlier ones, suffer from poor highlight line distributions near the singularities.…”

“…This proves tangent continuity at and near the irregularity. [8] presents a singularly-parameterized construction with four free control points per quad. However, the highlight line distribution of this simple uniformly-refinable C 1 construction are often pinched or otherwise irregular near the singularity, indicating insufficient quality for use as outer surfaces.…”

Isogeometric analysis with C1 hierarchical functions on planar two-patch geometries

“…vertices with valencies different to four), where the parameterization is singular, see e.g. [43,48,49], or consists of a special construction, see e.g. [33,34,42].…”