Generalized spline spaces over T-meshes: Dimension formula and locally refined generalized B-splines

Bracco, Cesare; Lyche, Tom; Manni, Carla; Roman, Fabio; Speleers, Hendrik

doi:10.1016/j.amc.2015.08.019

Cited by 17 publications

(33 citation statements)

References 36 publications

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“…{p} n−p ) the Bernstein basis relative to (a, b). One step of dimension diminishing transforms each Bernstein basis into the next one via (7) and (5). These relations can be read in the reverse way, which yields, for each p = n, n − 1, .…”

Section: Ec-spaces and Dimension Diminishingmentioning

confidence: 99%

“…We can then avoid any reference to weight functions, starting the recurrence formulae (8) from the Lagrange basis (β 0 , β 1 ) of E 1 . The same is generally done for the construction for splines [15,40,11,17,18,19,5].…”

Section: Critical Length: Why?mentioning

confidence: 99%

“…Cycloidal spaces are among the spaces which have also been extensively used during the last decade to build generalised splines (which are themselves examples of Chebyshevian or piecewise Chebyshevian splines) for Isogeometric Analysis purposes [11,17,18,19,5]. They belong to the larger class of spaces which are closed under reflection, corresponding to either odd or even characteristic polynomials.…”

Section: Introductionmentioning

confidence: 99%

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Critical length: An alternative approach

Beccari

Casciola

Mazure

2020

Journal of Computational and Applied Mathematics

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We provide a numerical method to determine the critical lengths of linear differential operators with constant real coefficients. The need for such a procedure arises when the orders increase. The interest of this article is clearly on the practical side since knowing the critical lengths permits an optimal use of the associated kernels. The efficiency of the procedure is due to its being based on crucial features of Extended Chebyshev spaces on closed bounded intervals.

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Section: Ec-spaces and Dimension Diminishingmentioning

confidence: 99%

Section: Critical Length: Why?mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Critical length: An alternative approach

Beccari

Casciola

Mazure

2020

Journal of Computational and Applied Mathematics

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“…Moreover, tensor-product GB-splines are an interesting problem-dependent alternative to tensor-product polynomial B-splines and NURBS in isogeometric analysis, a successful paradigm for the analysis of problems governed by partial differential equations (see [9,14,15]). Their success and the need of local refinement motivated the recent study of generalized splines over T-meshes (see [3,4,5,6]).…”

Section: Introductionmentioning

confidence: 99%

Numerical approximation of GB-splines by a convolutional approach

Roman

Manni

Speleers

2017

Applied Numerical Mathematics

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Generalized splines are smooth functions belonging piecewisely to spaces which are a natural generalization of algebraic polynomials. GB-splines are a B-spline-like basis for generalized splines, and they are usually defined by means of an integral recurrence relation which makes their evaluation quite cumbersome and computationally expensive. We present a simple strategy for approximating the values of a cardinal GB-spline of arbitrary degree p, with a particular focus on hyperbolic and trigonometric GB-splines due to their interest in applications. The proposed strategy is based on the Fourier properties of cardinal GB-splines. The approximant is expressed as a linear combination of scaled and dilated versions of (polynomial) cardinal B-splines of degree p, whose coefficients can be efficiently computed via discrete convolution. Sharp error estimates are provided and illustrated with some numerical examples.

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“…Tchebycheffian splines share many properties with the classical polynomial splines but also offer a much more flexible framework, due to the wide diversity of ET-spaces. Multivariate extensions of Tchebycheffian splines can be easily obtained via (local) tensor-product structures [7,8,9].…”

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confidence: 99%

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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Generalized spline spaces over T-meshes: Dimension formula and locally refined generalized B-splines

Cited by 17 publications

References 36 publications

Critical length: An alternative approach

Critical length: An alternative approach

Numerical approximation of GB-splines by a convolutional approach

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

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