Powell Sabin Triangulations Research Articles

An assessment of numerical and geometrical quality of bases on surface fitting on Powell–Sabin triangulations

It is well known that the problem of fitting a dataset by using a spline surface minimizing an energy functional can be carried out by solving a linear system. Such a linear system strongly depends on the underlying functional space and, particularly, on the basis considered. Some papers in the literature study the numerical behavior and processing of the above-mentioned linear systems in specific cases. The bases that have local support and constitute a partition of unity have been shown to be interesting in the frame of geometric problems. In this work, we investigate the numerical effects of considering these bases in the quadratic Powell–Sabin spline space. Specifically, we present a direct approach to explore different preconditioning strategies and assess whether the already known ‘good’ bases also possess favorable numerical properties. Additionally, we introduce an inverse optimization approach based on a nonlinear optimization model to identify new bases that exhibit both good geometric and numerical properties.

Convergence of Lagrange finite elements for the Maxwell eigenvalue problem in two dimensions

AbstractWe consider finite element approximations of the Maxwell eigenvalue problem in two dimensions. We prove, in certain settings, convergence of the discrete eigenvalues using Lagrange finite elements. In particular, we prove convergence in three scenarios: piecewise linear elements on Powell–Sabin triangulations, piecewise quadratic elements on Clough–Tocher triangulations and piecewise quartics (and higher) elements on general shape-regular triangulations. We provide numerical experiments that support the theoretical results. The computations also show that, on general triangulations, the eigenvalue approximations are very sensitive to nearly singular vertices, i.e., vertices that fall on exactly two ‘almost’ straight lines.

Quasi-Interpolation in a Space of C2 Sextic Splines over Powell–Sabin Triangulations

In this work, we study quasi-interpolation in a space of sextic splines defined over Powell–Sabin triangulations. These spline functions are of class C2 on the whole domain but fourth-order regularity is required at vertices and C3 regularity is imposed across the edges of the refined triangulation and also at the interior point chosen to define the refinement. An algorithm is proposed to define the Powell–Sabin triangles with a small area and diameter needed to construct a normalized basis. Quasi-interpolation operators which reproduce sextic polynomials are constructed after deriving Marsden’s identity from a more explicit version of the control polynomials introduced some years ago in the literature. Finally, some tests show the good performance of these operators.

A geometric characterization of Powell-Sabin triangulations allowing the construction of C2 quartic splines

The paper deals with the characterization of Powell-Sabin triangulations allowing the construction of bivariate quartic splines of class C2. The result is established by relating the triangle and edge split points provided by the refinement of each triangle. For a triangulation fulfilling the characterization obtained, a normalized representation of the splines in the C2 space is given.

Exact sequences on Powell–Sabin splits

We construct smooth finite elements spaces on Powell-Sabin triangulations that form an exact sequence. The first space of the sequence coincides with the classical $C^1$ Powell-Sabin space, while the others form stable and divergence-free yielding pairs for the Stokes problem. We develop degrees of freedom for these spaces that induce projections that commute with the differential operators.

Quartic splines on Powell–Sabin triangulations

The paper deals with the construction of bivariate quartic splines on Powell–Sabin triangulations. In particular, it provides a spline space that is C2 everywhere except across some edges of the refined triangulation. The splines that belong to this space can be described uniquely with interpolation values and derivatives of order at most two. Moreover, they can be represented in a locally supported basis that forms a convex partition of unity. As an application of this result, quasi-interpolants that reproduce polynomials of degree four are derived.

A normalized representation of super splines of arbitrary degree on Powell–Sabin triangulations

In the paper, a family of bivariate super spline spaces of arbitrary degree defined on a triangulation with Powell–Sabin refinement is introduced. It includes known spaces of arbitrary smoothness r and degree $$3r-1$$ but provides also other choices of spline degree for the same r which, in particular, generalize a known space of $$\mathscr {C}^{1}$$ cubic super splines. Minimal determining sets of the proposed super spline spaces of arbitrary degree are presented, and the interpolation problems that uniquely specify their elements are provided. Furthermore, a normalized representation of the discussed splines is considered. It is based on the definition of basis functions that have local supports, are nonnegative, and form a partition of unity. The basis functions share numerous similarities with classical univariate B-splines.

A new B-spline representation for cubic splines over Powell–Sabin triangulations

We consider a C1 cubic spline space defined over a triangulation with Powell–Sabin refinement. The space has some local C2 super-smoothness and can be seen as a close extension of the classical cubic Clough–Tocher spline space. In addition, we construct a suitable normalized B-spline representation for this spline space. The basis functions have a local support, they are nonnegative, and they form a partition of unity. We also show how to compute the Bézier control net of such a spline in a stable way.

A Family of Smooth Quasi-interpolants Defined Over Powell–Sabin Triangulations

We investigate the construction of local quasi-interpolation schemes based on a family of bivariate spline functions with smoothness \(r\ge 1\) and polynomial degree \(3r-1\). These splines are defined on triangulations with Powell–Sabin refinement, and they can be represented in terms of locally supported basis functions that form a convex partition of unity. With the aid of the blossoming technique, we first derive a Marsden-like identity representing polynomials of degree \(3r-1\) in such a spline form. Then we present a general recipe to construct various families of smooth quasi-interpolation schemes involving values and/or derivatives of a given function.

Preconditioned conjugate gradient method for finding minimal energy surfaces on Powell–Sabin triangulations

We present an iterative proposal, based on the preconditioned conjugate gradient method, to solve the linear system associated to the problem of approximating a data set by a minimal energy surface constructed through a Powell–Sabin finite element over a Δ1-type triangulation defined on a polygonal domain. These approximation problems give rise to symmetric, banded, and positive definite matrices with a very special block structure that depends on the basis functions of the associated vector space, and also on the numeration of the nodes in the triangulation. In practice, the associated sparse matrices are large and ill-conditioned. The special structure of these matrices allows us to adapt and explore several known preconditioned strategies to improve the performance of the conjugate gradient method.We adapt and explore five different preconditioning strategies, including some well-known direct and also some recent inverse strategies. Special attention is paid to the delicate and difficult task of choosing the related parameters in each case. The quality of each preconditioner is evaluated by observing the clustering of the preconditioned matrix eigenvalues, and the obtained reduction in number of iterations. We report on seven different surfaces, and our results indicate that the best preconditioning strategies for this application are the ones based on incomplete factorizations. Nevertheless, from a computational-cost point of view, all the explored strategies are competitive.

Construction of Normalized B-Splines for a Family of Smooth Spline Spaces Over Powell–Sabin Triangulations

We construct a suitable B-spline representation for a family of bivariate spline functions with smoothness r≥1 and polynomial degree 3r−1. They are defined on a triangulation with Powell–Sabin refinement. The basis functions have a local support, they are nonnegative, and they form a partition of unity. The construction involves the determination of triangles that must contain a specific set of points. We further consider a number of CAGD applications. We show how to define control points and control polynomials (of degree 2r−1), and we provide an efficient and stable computation of the Bernstein–Bézier form of such splines.

Bivariate C 2 cubic spline quasi-interpolants on uniform Powell–Sabin triangulations of a rectangular domain

In this paper we construct discrete quasi-interpolants based on C 2 cubic multi-box splines on uniform Powell–Sabin triangulations of a rectangular domain. The main problem consists in finding the coefficient functionals associated with boundary multi-box splines (i.e. multi-box splines whose supports overlap with the domain) involving data points inside or on the boundary of the domain and giving the optimal approximation order. They are obtained either by minimizing an upper bound for the infinity norm of the operator w.r.t. a finite number of free parameters, or by inducing the superconvergence of the gradient of the quasi-interpolant at some specific points of the domain. Finally, we give norm and error estimates and we provide some numerical examples illustrating the approximation properties of the proposed operators.

The C 1 wavelet method for numerical solutions of the Neumann problem

In this article we use the C 1 wavelet bases on Powell-Sabin triangulations to approximate the solution of the Neumann problem for partial differential equations. The C 1 wavelet bases are stable and have explicit expressions on a three-direction mesh. Consequently, we can approximate the solution of the Neumann problem accurately and stably. The convergence and error estimates of the numerical solutions are given. The computational results of a numerical example show that our wavelet method is well suitable to the Neumann boundary problem.

SURFACE COMPRESSION WITH HIERARCHICAL POWELL–SABIN B-SPLINES

We show how to construct a stable hierarchical basis for piecewise quadratic C1 continuous splines defined on Powell–Sabin triangulations. We prove that this hierarchical basis is well suited for compressing surfaces. Our compression method does not require the construction of wavelets which are usually expensive to compute, but instead we use a stable quasi-interpolation scheme that achieves optimal approximation order. Numerical experiments demonstrate the high compression rate of the algorithm.

Formula omitted] hierarchical Riesz bases of Lagrange type on Powell–Sabin triangulations

In this paper we construct C 1 continuous piecewise quadratic hierarchical bases on Powell–Sabin triangulations of arbitrary polygonal domains in R 2 . Our bases are of Lagrange type instead of the usual Hermite type and under some weak regularity assumptions on the underlying triangulations we prove that they form strongly stable Riesz bases for the Sobolev spaces H s ( Ω ) with s ∈ ( 1 , 5 / 2 ) . Especially the case s = 2 is of interest, because we can use the corresponding hierarchical basis for preconditioning fourth-order elliptic equations leading to uniformly well-conditioned stiffness matrices. Compared to the hierarchical Riesz bases by Davydov and Stevenson (Hierarchical Riesz bases for H s ( Ω ) , 1 < s < 5 / 2 . Constructive Approximation, to appear) our construction is simpler.

Macro-elements and stable local bases for splines on Powell-Sabin triangulations

Macro-elements of arbitrary smoothness are constructed on Powell-Sabin triangle splits. These elements are useful for solving boundary-value problems and for interpolation of Hermite data. It is shown that they are optimal with respect to spline degree, and we believe they are also optimal with respect to the number of degrees of freedom. The construction provides local bases for certain superspline spaces defined over Powell-Sabin refinements. These bases are shown to be stable as a function of the smallest angle in the triangulation, which in turn implies that the associated spline spaces have optimal order approximation power.