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On the linear independence of truncated hierarchical generating systems. (English) Zbl 1359.65027

Summary: Motivated by the necessity to perform adaptive refinement in geometric design and numerical simulation, the construction of hierarchical splines from generating systems spanning nested spaces has been recently studied in several publications. Linear independence can be guaranteed with the help of the local linear independence of the spline basis at each level. The present paper extends this framework in several ways. Firstly, we consider spline functions that are defined on domain manifolds, while the existing constructions are limited to domains that are open subsets of \(\mathbb{R}^d\). Secondly, we generalize the approach to generating systems containing functions which are not necessarily non-negative. Thirdly, we present a more general approach to guarantee linear independence and present a refinement algorithm that maintains this property. The three extensions of the framework are then used in several relevant applications: doubly hierarchical B-splines, hierarchical Zwart-Powell elements, and three different types of hierarchical subdivision splines.

MSC:

65D07 Numerical computation using splines
41A15 Spline approximation
65D17 Computer-aided design (modeling of curves and surfaces)

Software:

CHARMS; ISOGAT; IETI
PDFBibTeX XMLCite
Full Text: DOI

References:

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