Unstructured spline spaces for isogeometric analysis based on spline manifolds

TL;DR

This paper introduces a mathematical framework based on spline manifolds for unstructured B-spline spaces in isogeometric analysis, enabling analysis of complex multi-patch domains with extraordinary points.

Contribution

It generalizes dual-compatible B-splines to unstructured settings, ensuring key properties like linear independence and optimal approximation for $h$-refined meshes.

Findings

Framework supports analysis-suitable unstructured B-spline spaces

Proves linear independence and approximation properties

Applicable to complex multi-patch domains with extraordinary points

Abstract

Based on spline manifolds we introduce and study a mathematical framework for analysis-suitable unstructured B-spline spaces. In this setting the parameter domain has a manifold structure, which allows for the definition of function spaces that have a tensor-product structure locally, but not globally. This includes configurations such as B-splines over multi-patch domains with extraordinary points, analysis-suitable unstructured T-splines, or more general constructions. Within this framework, we generalize the concept of dual-compatible B-splines, which was originally developed for structured T-splines. This allows us to prove the key properties that are needed for isogeometric analysis, such as linear independence and optimal approximation properties for $h$-refined meshes.