$C^s$-smooth isogeometric spline spaces over planar multi-patch parameterizations

TL;DR

This paper develops a method to construct and analyze $C^s$-smooth isogeometric spline spaces over multi-patch geometries, extending previous work to arbitrary smoothness levels and demonstrating optimal approximation properties.

Contribution

It introduces a general framework for constructing $C^s$-smooth spline spaces over multi-patch domains for any smoothness $s \,\geq\, 1$, including basis construction and approximation analysis.

Findings

Constructed a basis of simple, locally supported functions for $C^s$-smooth spaces.

Demonstrated optimal $L^2$ approximation power of the constructed spaces.

Extended existing methods to arbitrary smoothness levels for multi-patch geometries.

Abstract

The design of globally $C^s$-smooth ($s \geq 1$) isogeometric spline spaces over multi-patch geometries is a current and challenging topic of research in the framework of isogeometric analysis. In this work, we extend the recent methods [25,28] and [31-33] for the construction of $C^1$-smooth and $C^2$-smooth isogeometric spline spaces over particular planar multi-patch geometries to the case of $C^s$-smooth isogeometric multi-patch spline spaces of an arbitrary selected smoothness $s \geq 1$. More precisely, for any $s \geq 1$, we study the space of $C^s$-smooth isogeometric spline functions defined on planar, bilinearly parameterized multi-patch domains, and generate a particular $C^s$-smooth subspace of the entire $C^s$-smooth isogeometric multi-patch spline space. We further present the construction of a basis for this $C^s$-smooth subspace, which consists of simple and locally supported functions. Moreover, we use the $C^s$-smooth spline functions to perform $L^2$ approximation on bilinearly parameterized multi-patch domains, where the obtained numerical results indicate an optimal approximation power of the constructed $C^s$-smooth subspace.