TDHB-splines: The truncated decoupled basis of hierarchical tensor-product splines

We consider the adaptive refinement of bivariate quartic C 2-smooth box-spline spaces on the three-directional (type-I) grid G. The polynomial segments of these box splines belong to a certain subspace of the space of quartic polynomials, which will be called the space of special quartics. Given a finite sequence (G ℓ) ℓ=0,...,N of dyadically refined grids, we obtain a hierarchical grid by selecting cells from each level such that their closure covers the entire domain Ω, which is a bounded subset of R 2. A suitable selection procedure allows to define a basis spanning a hierarchical box spline space. As our main result, we derive a characterization of this space. More precisely, under certain mild assumptions on hierarchical grid, the hierarchical spline space is shown to contain all C 2-smooth functions whose restrictions to the cells of the hierarchical grid are special quartic polynomials.

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“…This approach can be extended to the box spline case as well. Finally it is also possible to combine truncation and decoupling as in [12].…”

Section: Resultsmentioning

confidence: 99%

Characterization of bivariate hierarchical quartic box splines on a three-directional grid

Villamizar

Mantzaflaris

Jüttler

2016

Computer Aided Geometric Design

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“…In October 2013, Mokriš, Jüttler and Giannelli published the preprint [13] where they extended the results of [9] and, in addition, provided new insight for some of the results from [10]. We note that Theorem 1 can be alternatively obtained as a consequence of Theorem 3.5 [13]; it follows from the observation that the restriction on the configuration of domains given in Theorem 3.5 [13] is weaker than that of in Theorem 1.…”

Section: Addendummentioning

confidence: 80%

Bases of T-meshes and the refinement of hierarchical B-splines

Berdinsky

Kim

Cho

et al. 2015

Computer Methods in Applied Mechanics and Engineering

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Abstract. In this paper we consider spaces of bivariate splines of bi-degree (m, n) with maximal order of smoothness over domains associated to a two-dimensional grid. We define admissible classes of domains for which suitable combinatorial technique allows us to obtain the dimension of such spline spaces and the number of tensor-product B-splines acting effectively on these domains. Following the strategy introduced recently by Giannelli and Jüttler, these results enable us to prove that under certain assumptions about the configuration of a hierarchical T-mesh the hierarchical B-splines form a basis of bivariate splines of bi-degree (m, n) with maximal order of smoothness over this hierarchical T-mesh. In addition, we derive a sufficient condition about the configuration of a hierarchical T-mesh that ensures a weighted partition of unity property for hierarchical B-splines with only positive weights.

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“…Because hierarchical B-splines are generated directly from the T-meshes rather than the spline spaces, the completeness can not be guaranteed generally. Therefore, several articles [3,4,15,22,23] were published to discuss this question.…”

Section: Introductionmentioning

confidence: 99%