Dimension and basis construction for $C^{2}$-smooth isogeometric spline spaces over bilinear-like $G^{2}$ two-patch parameterizations

TL;DR

This paper develops a new basis construction for $C^{2}$-smooth isogeometric spline spaces over a broad class of bilinear-like $G^{2}$ two-patch geometries, enabling efficient approximation with good numerical properties.

Contribution

It introduces a novel basis for $C^{2}$-smooth isogeometric functions over bilinear-like $G^{2}$ two-patch geometries, including a uniform basis for all configurations.

Findings

The basis functions have simple closed forms and small supports.

The subspace achieves optimal approximation properties.

The dimension of the space is explicitly computed.

Abstract

A particular class of planar two-patch geometries, called bilinear-like $G^{2}$ two-patch geometries, is introduced. This class includes the subclass of all bilinear two-patch parameterizations and possesses similar connectivity functions along the patch interface. It is demonstrated that the class of bilinear-like $G^2$ two-patch parameterizations is much wider than the class of bilinear parameterizations and can approximate with good quality given generic two-patch parameterizations. We investigate the space of $C^{2}$-smooth isogeometric functions over this specific class of two-patch geometries. The study is based on the equivalence of the $C^2$-smoothness of an isogeometric function and the $G^2$-smoothness of its graph surface (cf. [12, 20]). The dimension of the space is computed and an explicit basis construction is presented. The resulting basis functions possess simple closed form representations, have small local supports, and are well-conditioned. In addition, we introduce a subspace whose basis functions can be generated uniformly for all possible configurations of bilinear-like $G^{2}$ two-patch parameterizations. Numerical results obtained by performing $L^{2}$-approximation indicate that already the subspace possesses optimal approximation properties.