Automatic construction of control triangles for subdivided Powell–Sabin splines

doi:10.1016/j.cagd.2004.05.001

Computer Aided Geometric Design

Volume 21, Issue 7, September 2004, Pages 671-682

https://doi.org/10.1016/j.cagd.2004.05.001 Get rights and content

Abstract

In this paper we present an algorithm for calculating the B-spline representation of a Powell–Sabin spline surface on a refinement of the given triangulation. The resulting subdivision scheme is a $3$ scheme; a new vertex is added inside every original triangle. Applying the $3$ scheme twice yields a triadic scheme, every original edge is split into three new edges, but special care is needed at the boundaries. The scheme is numerically stable and generally applicable, there are no restrictions on the initial triangulation.

References (10)

P. Dierckx
On calculating normalized Powell–Sabin B-splines
Computer Aided Geometric Design
(1997)
J. Windmolders et al.
Subdivision of Uniform Powell–Sabin splines
Computer Aided Geometric Design
(1999)
G. Farin
Triangular Bernstein–Bézier patches
Computer Aided Geometric Design
(1986)
L. Kobbelt
$3$ -subdivision
U. Labsik et al.
Interpolatory $3$ -subdivision

There are more references available in the full text version of this article.

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Generalized C1 Clough–Tocher splines for CAGD and FEM
2022, Computer Methods in Applied Mechanics and Engineering
The paper generalizes the classical $C^{1}$ cubic Clough–Tocher spline space over a triangulation to $C^{1}$ spaces of any degree higher that three. It shows that the considered spaces can be equipped with a basis consisting of non-negative locally supported functions forming a partition of unity and demonstrates the applicability of the basis in the context of the finite element method. The studied spaces have optimal approximation power and are defined by enforcing additional smoothness inside the triangles of the triangulation where the Clough–Tocher splitting is used. Locally, over each triangle of the triangulation, the splines are expressed in the Bernstein–Bézier form, which enables one to take the full advantage of the geometric properties and computational techniques that come with such a representation. Solving boundary problems with Galerkin discretization is thus relatively straightforward and is illustrated with several examples.
A B-spline basis for C1 quadratic splines on triangulations with a 10-split
2018, Journal of Computational and Applied Mathematics
The paper considers the macro-element splitting technique that refines every triangle of the initial triangulation into ten smaller triangles. The resulting refinement is an extension of the well-known Powell–Sabin 6-split and enables a construction of polynomial $C^{1}$ splines of degree two interpolating first order Hermite data at the vertices of the initial triangulation. A particular construction, called a balanced 10-split, is presented that allows a numerically stable B-spline representation of such splines. This amounts to, firstly, defining locally supported basis functions for the macro-element space that form a convex partition of unity, and, secondly, expressing the coefficients of the spline represented in this basis by the means of spline values and derivatives at the vertices of the initial triangulation.
Construction and analysis of cubic Powell–Sabin B-splines
2017, Computer Aided Geometric Design
Citation Excerpt :
As the B-spline coefficients naturally support a control structure, the representation might be a useful tool for geometric modelling. Thanks to the absence of local super-smoothness, it is possible to construct sequences of nested spline spaces and this allows the development of subdivision schemes in the same spirit as Vanraes et al. (2004); Windmolders and Dierckx (1999). The representation is also suited for the construction of quasi-interpolation schemes, following an approach similar to Manni and Sablonnière (2007); Sbibih et al. (2009).
We consider a new B-spline representation for the space of $C^{1}$ cubic splines defined on a triangulation with a Powell–Sabin refinement. The construction is based on lifting particular triangles and line segments from the domain. We prove that the B-splines form a locally supported stable basis and a convex partition of unity. Furthermore, we provide explicit expressions for the B-spline coefficients of any element of the cubic spline space and show how to compute the Bernstein–Bézier form of such a spline in a stable way. The B-spline representation induces a natural control structure that is useful for geometric modelling. Finally, we explore how classical $C^{1}$ quadratic Powell–Sabin splines and $C^{1}$ cubic Clough–Tocher splines can be expressed in the new B-spline representation.
C1 cubic splines on Powell-Sabin triangulations
2016, Applied Mathematics and Computation
A bivariate $C^{1}$ cubic spline space on a triangulation with Powell–Sabin refinement which extends the well-known $C^{1}$ quadratic spline space and has a nested structure is introduced. A construction of a locally supported basis that forms a partition of unity is presented based on choosing particular triangles and line segments in the domain. Further, it is shown how these objects can be determined in order to obtain nonnegative basis functions under a natural restriction on the Powell–Sabin refinement. Geometrically intuitive B-spline representation is proposed which makes these splines a useful tool for CAGD applications.
Superconvergent quadratic spline quasi-interpolants on Powell-Sabin partitions
2015, Applied Numerical Mathematics
In this paper we use Normalized Powell–Sabin B-splines constructed by Dierckx [6] to introduce a new B-spline representation of Hermite Powell–Sabin interpolant of any polynomial or any piecewise polynomial over Powell–Sabin partitions of class at least $C^{1}$ in terms of their polar forms. We use this representation for constructing several superconvergent discrete quasi-interpolants. The result that we present in this paper is a generalization of the one presented in [20] with other properties.
Powell-Sabin spline based multilevel preconditioners for the biharmonic equation
2010, Applied Numerical Mathematics
The Powell–Sabin (PS) piecewise quadratic $C^{1}$ finite element on the PS 12-split of a triangulation is a common choice for the construction of a BPX-type preconditioner for the biharmonic equation. In this note we investigate the related Powell–Sabin element on the PS 6-split instead of the PS 12-split for the construction of such preconditioners. For the PS 6-split element multilevel spaces can be created using a $\sqrt{3}$ -refinement scheme instead of the traditional dyadic scheme. Topologically $\sqrt{3}$ -refinement has many advantages: it is a slower refinement than the dyadic split operation, and it alternates the orientation of the refined triangles. Therefore we expect a reduction of the amount of work when compared to the PS 12-split element BPX preconditioner, although both methods have the same asymptotical complexity. Numerical experiments confirm this statement.

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Computer Aided Geometric Design

Abstract

References (10)

On calculating normalized Powell–Sabin B-splines

Computer Aided Geometric Design

Subdivision of Uniform Powell–Sabin splines

Computer Aided Geometric Design

Triangular Bernstein–Bézier patches

Computer Aided Geometric Design

$3$ -subdivision

Interpolatory $3$ -subdivision

Cited by (24)

Generalized C<sup>1</sup> Clough–Tocher splines for CAGD and FEM

A B-spline basis for C<sup>1</sup> quadratic splines on triangulations with a 10-split

Construction and analysis of cubic Powell–Sabin B-splines

C<sup>1</sup> cubic splines on Powell-Sabin triangulations

Superconvergent quadratic spline quasi-interpolants on Powell-Sabin partitions

Powell-Sabin spline based multilevel preconditioners for the biharmonic equation

Automatic construction of control triangles for subdivided Powell–Sabin splines

Abstract

Computer Aided Geometric Design

Computer Aided Geometric Design

Triangular Bernstein–Bézier patches

Computer Aided Geometric Design

3-subdivision

Interpolatory 3-subdivision

$3$ -subdivision

Interpolatory $3$ -subdivision