Spline approximation by quasiinterpolants

Boor, Carl de; Fix, George J.

doi:10.1016/0021-9045(73)90029-4

Cited by 301 publications

(146 citation statements)

References 9 publications

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“…Equation (8) and N 2q−1 of degree 2p − 1 and 2q − 1 with the knot vectors obtained from Ξ u and Ξ v by raising the multiplicities of the inner u-knots and v-knots by p and q, respectively [24]. That is,…”

Section: Validating a B-spline Parametrizationmentioning

confidence: 99%

Isogeometric Shape Optimization for Electromagnetic Scattering Problems

Nguyen¹,

Evgrafov

Gravesen³

2012

PIER B

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Abstract-We consider the benchmark problem of magnetic energy density enhancement in a small spatial region by varying the shape of two symmetric conducting scatterers. We view this problem as a prototype for a wide variety of geometric design problems in electromagnetic applications. Our approach for solving this problem is based on shape optimization and isogeometric analysis. One of the major difficulties we face to make these methods work together is the need to maintain a valid parametrization of the computational domain during the optimization. Our approach to generating a domain parametrization is based on minimizing a second order approximation to the Winslow functional in the vicinity of a reference parametrization. Furthermore, we enforce the validity of the parametrization by ensuring the non-negativity of the coefficients of a B-spline expansion of the Jacobian. The shape found by this approach outperforms earlier design computed using topology optimization by a factor of one billion.

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Section: Validating a B-spline Parametrizationmentioning

confidence: 99%

Isogeometric Shape Optimization for Electromagnetic Scattering Problems

Nguyen¹,

Evgrafov

Gravesen³

2012

PIER B

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“…To ensure good approximation properties it is important that the methods reproduce polynomials and preferably the functions in the given spline space. A method based on derivative information was constructed in [3], while a more general class was studied in [9]. In order to reproduce the spline space, the local information of the methods in [9] was restricted to lie in one knot interval.…”

Section: Quasi-interpolantsmentioning

confidence: 99%

“…The splines approximation technique used in this paper is quasi-interpolants, first developed by de Boor and Fix [3]. The quasi-interpolants operators were later generalized by Lyche and Schumaker [9], and it is their version that used in the alternative surface approximation technique.…”

Section: Introductionmentioning

confidence: 99%

Quasi-interpolants Based Multilevel B-Spline Surface Reconstruction from Scattered Data

Lee

Kwon

2005

Computational Science and Its Applications – ICCSA 2005

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Abstract. This paper presents a new fast and local method of 3D surface reconstruction for scattered data. The algorithm makes use of quasiinterpolants to compute the control points from a coarse to fine hierarchy to generate a sequence of bicubic B-spline functions whose sum approaches to the desired interpolation function. Quasi-interpolants gives a procedure for deriving local spline approximation methods where a Bspline coefficient only depends on data points taken from the neighborhood of the support corresponding B-spline. Experimental results demonstrate that high-fidelity reconstruction is possible from a selected set of irregular samples.

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“…Another extension consists in adding a Jacobi weight in the scalar product. The prototypes of spline DQIs are de Boor-Fix QIs [11] and their various univariate and multivariate extensions…”

Section: Introductionmentioning

confidence: 99%

Recent Progress on Univariate and Multivariate Polynomial and Spline Quasi-interpolants

Sablonnière

2005

Trends and Applications in Constructive Approximation

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Polynomial and spline quasi-interpolants (QIs) are practical and effective approximation operators. Among their remarkable properties, let us cite for example: good shape properties, easy computation and evaluation (no linear system to solve), uniform boundedness independently of the degree (polynomials) or of the partition (splines), good approximation order. We shall emphasize new results on various types of univariate and multivariate polynomial or spline QIs, depending on the nature of coefficient functionals, which can be differential, discrete or integral. We shall also present some applications of QIs to numerical methods.

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Spline approximation by quasiinterpolants

Cited by 301 publications

References 9 publications

Isogeometric Shape Optimization for Electromagnetic Scattering Problems

Isogeometric Shape Optimization for Electromagnetic Scattering Problems

Quasi-interpolants Based Multilevel B-Spline Surface Reconstruction from Scattered Data

Recent Progress on Univariate and Multivariate Polynomial and Spline Quasi-interpolants

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