Efficient quadrature rules based on spline quasi-interpolation for application to IGA-BEMs

Calabrò, Francesco; Falini, Antonella; Sampoli, Maria Lucia; Sestini, Alessandra

doi:10.1016/j.cam.2018.02.005

Cited by 30 publications

(49 citation statements)

References 20 publications

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“…Thus, we obtain three piece-wise polynomial equations which are cubic in u 1 , v 2 , and w 0 ; cf. (7). Using a computer algebra system such as Maple, it can be shown that this system, in all its eight polynomial variants based on the ranges of u 1 , v 2 , and w 0 , admits only one solution: u 1 = v 2 = w 0 = 1/2, as we set out to prove.…”

Section: Gaussian Quadrature For C 1 Quadratic Powell-sabin Macro-trimentioning

confidence: 92%

“…Let the split-points be defined as in (7). Since dim(S 1 2 (T )) = 9 and dim(Π 2 ) = 6, we first define three spline basis functions that expand the basis of Π 2 to a basis of S 1 2 .…”

Section: Gaussian Quadrature For C 1 Quadratic Powell-sabin Macro-trimentioning

confidence: 99%

“…Typically, the rule is required to be exact, that is, R m (f ) ≡ 0 for each element of a predefined linear function space L. Moreover, the rule is said to be Gaussian if m is the minimal number of nodes t i ∈ R n at which f has to be evaluated. There is an extensive number of various quadrature rules depending on n (f is univariate [15], bivariate [28,39], multivariate [16]), domain shape (disc, hypercube, simplex) [37], and the type of the linear space (polynomials [15], splines [4,6,29,33], rational functions [38], smooth non-polynomial [7,27]). For polynomial multivariate integration, the field is well studied and the reader is referred to [37].…”

Section: Introductionmentioning

confidence: 99%

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On numerical quadrature for C1 quadratic Powell–Sabin 6-split macro-triangles

Bartoň

Kosinka

2019

Journal of Computational and Applied Mathematics

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The quadrature rule of Hammer and Stroud [16] for cubic polynomials has been shown to be exact for a larger space of functions, namely the C 1 cubic Clough-Tocher spline space over a macro-triangle if and only if the split-point is the barycentre of the macro-triangle [21]. We continue the study of quadrature rules for spline spaces over macro-triangles, now focusing on the case of C 1 quadratic Powell-Sabin 6-split macro-triangles. We show that the 3-node Gaussian quadrature(s) for quadratics can be generalised to the C 1 quadratic Powell-Sabin 6-split spline space over a macro-triangle for a two-parameter family of inner split-points, not just the barycentre as in [21]. The choice of the inner split-point uniquely determines the positions of the edge split-points such that the whole spline space is integrated exactly by a corresponding polynomial quadrature. Consequently, the number of quadrature points needed to exactly integrate this special spline space reduces from twelve to three.For the inner split-point at the barycentre, we prove that the two 3-node quadratic polynomial quadratures of Hammer and Stroud exactly integrate also the C 1 quadratic Powell-Sabin spline space if and only if the edge split-points are at their respective edge midpoints. For other positions of the inner and edge split-points we provide numerical examples showing that three nodes suffice to integrate the space exactly, but a full classification and a closed-form solution in the generic case remain elusive.

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Section: Gaussian Quadrature For C 1 Quadratic Powell-sabin Macro-trimentioning

confidence: 92%

“…Let the split-points be defined as in (7). Since dim(S 1 2 (T )) = 9 and dim(Π 2 ) = 6, we first define three spline basis functions that expand the basis of Π 2 to a basis of S 1 2 .…”

Section: Gaussian Quadrature For C 1 Quadratic Powell-sabin Macro-trimentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

On numerical quadrature for C1 quadratic Powell–Sabin 6-split macro-triangles

Bartoň

Kosinka

2019

Journal of Computational and Applied Mathematics

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“…The quadrature rules adopted here are based on a QI operator firstly introduced in the work of Mazzia and Sestini and applied to nonsingular numerical integration in the other work of Mazzia and Sestini . Then, the same QI‐based idea was adopted in the work of Calabrò et al to develop efficient and competitive quadratures for singular integrals appearing in the IgA‐BEM context. The rules are effective also for nearly singular integrals.…”

Section: Introductionmentioning

confidence: 99%

“…In this paper, we present a new adaptive IgA‐BEM with hierarchical B‐splines based on local quadrature schemes to solve 2D Laplace problems. In particular, the recently developed spline QI‐based quadrature schemes, derived in the works of Mazzia and Sestini and Calabrò et al for regular and singular integrals, respectively, are here investigated in the context of adaptive BEMs and properly framed in the hierarchical setting. The considered quadrature technique is based on isolating the singular part of the integral from the geometrical contribution and then by separately integrating the simplified singular integral.…”

Section: Introductionmentioning

confidence: 99%

An adaptive IgA‐BEM with hierarchical B‐splines based on quasi‐interpolation quadrature schemes

Falini

Giannelli

Kanduč

et al. 2018

Numerical Meth Engineering

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Summary The isogeometric formulation of the boundary element method (IgA‐BEM) is investigated within the adaptivity framework. Suitable weighted quadrature rules to evaluate integrals appearing in the Galerkin BEM formulation of 2D Laplace model problems are introduced. The proposed quadrature schemes are based on a spline quasi‐interpolation (QI) operator and properly framed in the hierarchical setting. The local nature of the QI perfectly fits with hierarchical spline constructions and leads to an efficient and accurate numerical scheme. An automatic adaptive refinement strategy is driven by a residual‐based error estimator. Numerical examples show that the optimal convergence rate of the Galerkin solution is recovered by the proposed adaptive method.

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