On numerical quadrature for \(C^1\) quadratic Powell-Sabin 6-split macro-triangles
type: Item
instance of: scholarly article
title: On numerical quadrature for \(C^1\) quadratic Powell-Sabin 6-split macro-triangles
zbMATH Open document ID: 1455.65026
author: Michael Bartoň
author: Jiří Kosinka
published in: Journal of Computational and Applied Mathematics
publication date: 2019-01-24T00:00:00Z
review text: In this paper the authors study Gaussian quadrature rules for polynomial spline spaces over triangulations, in particular for the case of $C^1$ quadratic Powell-Sabin 6-split macro-triangles, often used in interpolation problems over triangulations and in isogeometric analysis. After an introduction on Bézier triangles, Powell-Sabin 6-split and Hammer-Stroud quadratures for quadratic triangular polynomials is presented. Such quadratures ensure that the number of nodes is reduced from twelve or eighteen to three (optimal number). The construction of three new spline basis functions complements the basis of bivariate quadratic polynomials on a triangle to form a basis over the Powell-Sabin macro-element. The authors also generalize the problem to the non-symmetric case, i.e., to quadrature rules for the Powell-Sabin splines with general split-points. They obtain numerical evidence that three nodes suffice also in the general situation, but a rigorous proof of this fact is yet to be shown.
reviewed by: Paola Lamberti
Mathematics Subject Classification ID: 65D30
Mathematics Subject Classification ID: 65D07
Mathematics Subject Classification ID: 65D32
zbMATH DE Number: 7006426
zbMATH Keywords: numerical integration
zbMATH Keywords: Powell-Sabin spline space
zbMATH Keywords: Gaussian quadrature rules
MaRDI profile type: Publication
full work available at URL: j.cam.2018.07.051
OpenAlex ID: W2886172065
cites work: Q3231529
cites work: Calculation of Gauss Quadrature Rules
cites work: Generalized Gaussian quadrature rules on arbitrary polygons
cites work: A numerical algorithm for the construction of efficient quadrature rules in two and higher dimensions
cites work: Numerical Integration Over Simplexes
cites work: Q4158394
cites work: A quadrature-based approach to improving the collocation method
cites work: Q4847717
cites work: Gaussian quadrature rules for \(C^1\) quintic splines with uniform knot vectors
cites work: On computing rational Gauss-Chebyshev quadrature formulas
cites work: Generalized Gaussian Quadrature Rules for Systems of Arbitrary Functions
cites work: Efficient quadrature rules based on spline quasi-interpolation for application to IGA-BEMs
cites work: Isogeometric Analysis
cites work: Efficient quadrature for NURBS-based isogeometric analysis
cites work: A simple algorithm for obtaining nearly optimal quadrature rules for NURBS-based isogeometric analysis
cites work: Fast formation of isogeometric Galerkin matrices by weighted quadrature
cites work: Gaussian quadrature for splines via homotopy continuation: rules for \(C^2\) cubic splines
cites work: Optimal and reduced quadrature rules for tensor product and hierarchically refined splines in isogeometric analysis
cites work: Optimal quadrature for univariate and tensor product splines
cites work: Box splines
cites work: Q5535928
cites work: Numerical integration of 2-D integrals based on local bivariate \(C^1\) quasi-interpolating splines
cites work: Numerical integration based on bivariate quadratic spline quasi-interpolants on bounded domains
cites work: Numerical integration based on bivariate quadratic spline quasi-interpolants on Powell-Sabin partitions
cites work: Q4325916
cites work: Gaussian quadrature for \(C^1\) cubic Clough-Tocher macro-triangles
cites work: Piecewise Quadratic Approximations on Triangles
cites work: Spline Functions on Triangulations
cites work: Nonnegativity preserving macro-element interpolation of scattered data
cites work: Watertight conversion of trimmed CAD surfaces to Clough-Tocher splines
cites work: Macro-elements and stable local bases for splines on Clough-Tocher triangulations
cites work: A normalized basis for reduced Clough-Tocher splines
cites work: Smooth macro-elements based on Clough-Tocher triangle splits
cites work: Multiresolution analysis over triangles, based on quadratic Hermite interpolation
cites work: Surfaces in computer aided geometric design: A survey with new results
cites work: Optimizing domain parameterization in isogeometric analysis based on Powell-Sabin splines
DOI: 10.1016/J.CAM.2018.07.051
description: scientific article; zbMATH DE number 7006426