THB-splines: The truncated basis for hierarchical splines

The construction of classical hierarchical B-splines can be suitably modified in order to define locally supported basis functions that form a partition of unity. We will show that this property can be obtained by reducing the support of basis functions defined on coarse grids, according to finer le...

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Bibliographic Details
Published in:Computer aided geometric design Vol. 29; no. 7; pp. 485 - 498
Main Authors: Giannelli, Carlotta, Jüttler, Bert, Speleers, Hendrik
Format: Journal Article Conference Proceeding
Language:English
Published: Kidlington Elsevier B.V 01.10.2012
Elsevier
Subjects:
Exact sciences and technology
Hierarchical tensor–product B-splines
Local refinement
Mathematics
Methods of scientific computing (including symbolic computation, algebraic computation)
Numerical analysis
Numerical analysis. Scientific computation
Numerical approximation
Partition of unity
Sciences and techniques of general use
Truncated basis
Truncated basis
Partition of unity
Hierarchical tensor–product B-splines
Local refinement
B spline
Refinement method
Tensor product
Grid
Refinement
Adaptive method
Hierarchized structure
Spline approximation
Truncation
Hierarchy
Truncated shape
Hierarchical tensor-product B-splines
ISSN:0167-8396, 1879-2332
Online Access:Get full text
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Summary:The construction of classical hierarchical B-splines can be suitably modified in order to define locally supported basis functions that form a partition of unity. We will show that this property can be obtained by reducing the support of basis functions defined on coarse grids, according to finer levels in the hierarchy of splines. This truncation not only decreases the overlapping of supports related to basis functions arising from different hierarchical levels, but it also improves the numerical properties of the corresponding hierarchical basis — which is denoted as truncated hierarchical B-spline (THB-spline) basis. Several computed examples will illustrate the adaptive approximation behavior obtained by using a refinement algorithm based on THB-splines. ► A normalized hierarchical tensor–product B-spline basis is proposed. ► The construction relies on a truncation mechanism of coarse basis functions. ► The local refinement property is illustrated by an adaptive approximation strategy. ► Sparsity and condition numbers of the related matrices are studied experimentally.
ISSN:0167-8396
1879-2332
DOI:10.1016/j.cagd.2012.03.025