Construction of Smooth Isogeometric Function Spaces on Singularly Parameterized Domains

TL;DR

This paper develops a method to construct smooth isogeometric function spaces on domains with reduced regularity by locally mapping singular patches to triangular domains, enabling prescribed smoothness.

Contribution

It introduces a novel local mapping technique for singular tensor-product patches to triangular domains, facilitating smooth isogeometric functions on irregular geometries.

Findings

Constructed smooth basis functions on singular domains.

Mapped singular patches onto triangular domains for regularity.

Discussed potential for higher-dimensional generalizations.

Abstract

We aim at constructing a smooth basis for isogeometric function spaces on domains of reduced geometric regularity. In this context an isogeometric function is the composition of a piecewise rational function with the inverse of a piecewise rational geometry parameterization. We consider two types of singular parameterizations, domains where a part of the boundary is mapped onto one point and domains where parameter lines are mapped collinearly at the boundary. We locally map a singular tensor-product patch of arbitrary degree onto a triangular patch, thus splitting the parameterization into a singular bilinear mapping and a regular mapping on a triangular domain. This construction yields an isogeometric function space of prescribed smoothness. Generalizations to higher dimensions are also possible and are briefly discussed in the final section.