A basis of multi-degree splines

Shen, Wanqiang; Wang, Guozhao

doi:10.1016/j.cagd.2009.08.005

Cited by 22 publications

(20 citation statements)

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“…4-Nevertheless, if higher dimensions are needed, the richness of the present context can be efficiently exploited using splines obtained by (repeated) integration. For instance, through Proposition 7.5 this provides access to examples of splines visually identical to some so-called multi-degree splines (see, e.g., [46,47]), with more regularity. …”

Section: Concluding Commentsmentioning

confidence: 99%

Constructing totally positive piecewise Chebyshevian B-spline bases

Mazure

2018

Journal of Computational and Applied Mathematics

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Section: Concluding Commentsmentioning

confidence: 99%

Constructing totally positive piecewise Chebyshevian B-spline bases

Mazure

2018

Journal of Computational and Applied Mathematics

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“…Multi-degree splines were introduced in [34] for degrees (1,2,3) and (1, n). In a more general framework, basis functions for multi-degree splines were constructed in [37,36,38] by means of recursive integral relations involving global quantities that [19] observed as being difficult to compute. Instead, [19] presented a recursive geometric algorithm for computing multi-degree spline curves.…”

Section: Related Literaturementioning

confidence: 99%

Multi-degree smooth polar splines: A framework for geometric modeling and isogeometric analysis

Toshniwal

Speleers

Hiemstra

et al. 2017

Computer Methods in Applied Mechanics and Engineering

101

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We develop a multi-degree polar spline framework with applications to both geometric modeling and isogeometric analysis. First, multi-degree splines are introduced as piecewise non-uniform rational B-splines (NURBS) of non-uniform or variable polynomial degree, and a simple algorithm for their construction is presented. Then, an extension to two-dimensional polar configurations is provided by means of a tensor-product construction with a collapsed edge. Suitable combinations of these basis functions yield C k smooth polar splines for any k ≥ 0. We show that it is always possible to construct a set of smooth polar spline basis functions that form a convex partition of unity and possess locality. Explicit constructions for k ∈ {0, 1, 2} are presented. Optimal approximation behavior is observed numerically, and examples of free-form design, smooth hole-filling, and high-order partial differential equations demonstrate the applicability of the developed framework.

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“…The above contribution is complemented by additional results: we provide a knot insertion formula and a global integral recurrence relation for GTB-splines. While the former is in complete analogy with the one known for the multi-degree polynomial case [2,42], the latter is a new contribution also for the multi-degree polynomial case, where only local integral recurrence relations have been proposed so far in the literature [2,38]. The provided global integral recurrence relation completely mimics the one known for polynomial/Tchebycheffian splines of uniform degree/local dimension and is expressed in an elegant way by using an extension of the concept of weights.…”

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confidence: 98%

“…1 2 R. R. HIEMSTRA, T. J. R. HUGHES, C. MANNI, H. SPELEERS, AND D. TOSHNIWAL spect to differentiation and integration makes them an appealing substitute for the rational NURBS model in the framework of both Galerkin and collocation isogeometric methods [1,23,25,26]. When the geometry is not an issue, Tchebycheffian splines can still provide an interesting problem-dependent alternative to classical polynomial B-splines/NURBS for solving differential problems: they allow for an efficient treatment of sharp gradients and thin layers [24,25] and are able to outperform classical polynomial B-splines in the spectral approximation of differential operators [25,26].The success of polynomial splines greatly relies on the famous B-spline basis which can also be defined in the multi-degree setting [2,33,38,39,41]. Most of the results known for polynomial splines extend in a natural way to Tchebycheffian splines.…”

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confidence: 99%

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A Tchebycheffian Extension of Multidegree B-Splines: Algorithmic Computation and Properties

Hiemstra¹,

Hughes²,

Manni³

et al. 2020

SIAM J. Numer. Anal.

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In this paper we present an efficient and robust approach to compute a normalized Bspline-like basis for spline spaces with pieces drawn from extended Tchebycheff spaces. The extended Tchebycheff spaces and their dimensions are allowed to change from interval to interval. The approach works by constructing a matrix that maps a generalized Bernstein-like basis to the B-spline-like basis of interest. The B-spline-like basis shares many characterizing properties with classical univariate B-splines and may easily be incorporated in existing spline codes. This may contribute to the full exploitation of Tchebycheffian splines in applications, freeing them from the restricted role of an elegant theoretical extension of polynomial splines. Numerical examples are provided that illustrate the procedure described. 1 2 R. R. HIEMSTRA, T. J. R. HUGHES, C. MANNI, H. SPELEERS, AND D. TOSHNIWAL spect to differentiation and integration makes them an appealing substitute for the rational NURBS model in the framework of both Galerkin and collocation isogeometric methods [1,23,25,26]. When the geometry is not an issue, Tchebycheffian splines can still provide an interesting problem-dependent alternative to classical polynomial B-splines/NURBS for solving differential problems: they allow for an efficient treatment of sharp gradients and thin layers [24,25] and are able to outperform classical polynomial B-splines in the spectral approximation of differential operators [25,26].The success of polynomial splines greatly relies on the famous B-spline basis which can also be defined in the multi-degree setting [2,33,38,39,41]. Most of the results known for polynomial splines extend in a natural way to Tchebycheffian splines. However, the possibility of representing the space in terms of a basis with similar properties to polynomial B-splines is not always guaranteed, even for pieces taken from ET-spaces of the same dimension. More precisely, there are two main categories of Tchebycheffian splines: the various pieces are drawn either from the same ET-space -see [33] for a proper meaning in case of different local dimensions -or from different ET-spaces. In the former case, Tchebycheffian splines always admit a representation in terms of basis functions with similar properties to polynomial Bsplines. The latter offers a much more general framework -sometimes referred to in the literature as piecewise Tchebycheffian splines [31] -and allows us to optimally benefit from the great diversity of ET-spaces, but the existence of a B-spline-like basis requires constraints on the various ET-spaces.Tchebycheffian splines can be easily incorporated in existing spline codes because the corresponding B-spline-like basis, whenever it exists, is compatible with classical B-splines as it enjoys the same structural properties. When all the ET-spaces have the same dimension, various approaches have been used in the Tchebycheffian setting to construct such a B-spline-like basis: generalized divided differences [32,34], Hermite interpolation [10,33], integral...

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A basis of multi-degree splines

Cited by 22 publications

References 1 publication

Constructing totally positive piecewise Chebyshevian B-spline bases

Constructing totally positive piecewise Chebyshevian B-spline bases

Multi-degree smooth polar splines: A framework for geometric modeling and isogeometric analysis

A Tchebycheffian Extension of Multidegree B-Splines: Algorithmic Computation and Properties

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