Spline Spaces Research Articles

Dimension of polynomial splines of mixed smoothness on T-meshes

In this paper we study the dimension of splines of mixed smoothness on axis-aligned T-meshes. This is the setting when different orders of smoothness are required across the edges of the mesh. Given a spline space whose dimension is independent of its T-mesh's geometric embedding, we present constructive and sufficient conditions that ensure that the smoothness across a subset of the mesh edges can be reduced while maintaining stability of the dimension. The conditions have a simple geometric interpretation. Examples are presented to show the applicability of the results on both hierarchical and non-hierarchical T-meshes. For hierarchical T-meshes it is shown that mixed smoothness spline spaces that contain the space of PHT-splines (Deng et al., 2008) always have stable dimension.

A sharp degree bound on [formula omitted]-refinable multi-sided surfaces

Refinement of a space of splines should yield additional degrees of freedom for modeling and engineering analysis, both along boundaries and in the interior. Yet such additional flexibility fails to materialize for multi-sided G2 surface constructions when the polynomial degree is too low.This paper establishes a tight lower bound on the polynomial degree of flexibility-increasing refinable multi-sided G2 surface constructions within a C2 spline complex — by ruling out bi-5 constructions and by exhibiting a multi-sided bi-6 construction that yields good highlight line and curvature distributions. The bi-6 construction consists of one 2 × 2 macro-patch for each of the n sectors that join to form the multi-sided surface.

Local (T)HB-spline projectors via restricted hierarchical spline fitting

This paper is devoted to techniques for adaptive spline projection via quasi-interpolation, enabling the efficient approximation of given functions. We employ local least-squares fitting in restricted hierarchical spline spaces to establish novel projection operators for hierarchical splines of degree p. This leads to efficient spline projectors that require O(pd) floating point operations and O(1) evaluations of the given function per degree of freedom, while providing essentially the same accuracy as global approximation. Our spline projectors are based on a unifying framework for quasi-interpolation in hierarchical spline spaces. We present a detailed comparison with the scheme of Speleers and Manni (2016).

Fast formation of isogeometric Galerkin matrices via integration by interpolation and look-up

Although isogeometric analysis possesses many advantages over classical finite element methods, the computational costs of matrix assembly (especially for high polynomial degrees) constitute a bottleneck in isogeometric numerical simulations. To address this issue, we propose an efficient algorithm for the formation of isogeometric Galerkin matrices based on the interpolation, look-up and sum factorization techniques. This method consists of three steps: First, we project the common factors occurring in the integrals into an appropriate spline space via the interpolation or quasi-interpolation operator. Subsequently, the entries of the stiffness and mass matrices are approximated by a sum of integrals of tensor-product B-splines. Second, to perform an exact integration of the integrands in the approximated matrices, a look-up table for the standardized B-spline tri-product integrals is built. Finally, the system matrices are efficiently assembled by invoking the sum factorization technique. We present a detailed analysis of the computational costs, in order to compare the new method with the existing approaches for all polynomial degrees. Several numerical tests confirm that the proposed method ensures the efficiency of matrix assembly. In addition, the extension of our approach to matrix-free applications is also discussed.

Isogeometric analysis with [formula omitted] hierarchical functions on planar two-patch geometries

Adaptive isogeometric methods for the solution of partial differential equations rely on the construction of locally refinable spline spaces. A simple and efficient way to obtain these spaces is to apply the multi-level construction of hierarchical splines, that can be used on single-patch domains or in multi-patch domains with C0 continuity across the patch interfaces. Due to the benefits of higher continuity in isogeometric methods, recent works investigated the construction of spline spaces with global C1 continuity on two or more patches. In this paper, we show how these approaches can be combined with the hierarchical construction to obtain global C1 continuous hierarchical splines on two-patch domains. A selection of numerical examples is presented to highlight the features and effectivity of the construction.

Argyris type quasi-interpolation of optimal approximation order

The paper investigates a quasi-interpolation framework for the construction of operators that provide approximations for bivariate functions from the space of Argyris macro-element splines on a general triangulation. An operator of such type is determined by local approximation operators that produce quintic polynomials and are naturally associated with vertices and triangles of the triangulation. The quasi-interpolant resembles the properties of the standard Argyris interpolant, especially in the determination of data corresponding to the vertices of the triangulation. The main difference is in the definition of the cross-boundary derivatives, where a novel approach with degree raising is used to eliminate the need for additional interpolation of derivatives at the edges. Despite reduction in degrees of freedom, a suitable choice of local approximation operators ensures that the quasi-interpolation operator has quintic precision and is of optimal approximation order. To demonstrate the usability of the presented approach, three approximation methods are derived based on local Hermite and Lagrange interpolation and least square fitting.

Bivariate semialgebraic splines

Semialgebraic splines are bivariate splines over meshes whose edges are arcs of algebraic curves. They were first considered by Wang, Chui, and Stiller. We compute the dimension of the space of semialgebraic splines in two extreme cases. If the polynomials defining the edges span a three-dimensional space of polynomials, then we compute the dimensions from the dimensions for a corresponding rectilinear mesh. If the mesh is sufficiently generic, we give a formula for the dimension of the spline space valid in large degree and bound how large the degree must be for the formula to hold. We also study the dimension of the spline space in examples which do not satisfy either extreme. The results are derived using commutative and homological algebra.

Isogeometric Bézier dual mortaring: The enriched Bézier dual basis with application to second- and fourth-order problems

In this paper, we present an algorithm to construct enriched Bézier dual basis functions that can reproduce higher-order polynomials. Our construction is unique in that it is based on Bézier extraction and projection, allowing it to be used for tensor product and unstructured polynomial spline spaces, is well-conditioned, and is quadrature-free. When used as a basis for dual mortar methods, optimal approximations are achieved for both second- and fourth-order problems. In the context of fourth-order problems, both C0 and C1 continuity constraints must be applied at each intersection. We develop a novel geometry-independent C1 continuity constraint that also preserves the sparsity of the coupled problem. The performance of the proposed formulation is verified through several challenging second- and fourth-order problems.

Graphs admitting only constant splines

We study {\em generalized graph splines,} introduced by Gilbert, Viel, and the last author. For a large class of rings, we characterize the graphs that only admit constant splines. To do this, we prove that if a graph has a particular type of cutset (e.g., a bridge), then the space of splines naturally decomposes as a certain direct sum of submodules. As an application, we use these results to describe splines on a triangulation studied by Zhou and Lai, but over a different ring than they used.

A superconvergent nonconforming quadrilateral spline element for biharmonic equation using the B-net method

In this paper, we construct a new nonconforming quadrilateral element with 12 degrees of freedom to solve the biharmonic problems. $${\mathscr {T}}_{h}$$ is a triangulated quadrangulation of the domain $$\varOmega $$ . For a quadrilateral element $$Q_{T}$$ , the finite element space, which contains $${\mathbb {P}}_{3}(Q_{T})$$ , is a subspace of the bivariate spline space $${\mathbf {S}}_{3}^{1}(Q_{T})$$ . The degrees of freedom are chosen as the four point values, the four edge integrals of the shape functions and the edge integrals of their normal derivatives such that the weak continuity between elements can be satisfied. Accordingly, we explicitly establish 12 spline interpolation bases in the B-net form. Error estimates are given with optimal convergence order in both discrete $$H^{2}$$ and $$H^{1}$$ seminorms. The proposed element NCQS12 can get the superconvergence results with theoretical proof when $${\mathscr {T}}_{h}$$ is an uniform parallelogram mesh. Some degenerate meshes are considered subsequently. Finally, we do some numerical experiments to verify the theoretical analysis. Numerical results show that the proposed element performs well over the asymptotically regular parallelogram meshes, which is same as over the uniform parallelogram meshes.

Geometrically smooth spline bases for data fitting and simulation

Given a topological complex M with glueing data along edges shared by adjacent faces, we study the associated space of geometrically smooth spline functions that satisfy differentiability properties across shared edges. We present new and efficient constructions of basis functions of the space of G1-spline functions on quadrangular meshes, which are tensor product b-spline functions on each quadrangle and with b-spline transition maps across the shared edges. This new strategy for constructing basis functions is based on a local analysis of the edge functions, and does not depend on the global topology of M. We show that the separability of the space of G1 splines across an edge allows to determine the dimension and a basis of the space of G1 splines on M. This leads to explicit and effective constructions of basis functions attached to the vertices, edges and faces of M.This basis construction has important applications in geometric modeling and simulation. We illustrate it by the fitting of point clouds by G1 splines on quadrangular meshes of complex topology and in Isogeometric Analysis methods for the solution of diffusion equations. The ingredients are detailed and experimentation results showing the behavior of the method are presented.

Explicit error estimates for spline approximation of arbitrary smoothness in isogeometric analysis

In this paper we provide a priori error estimates with explicit constants for both the $L^2$-projection and the Ritz projection onto spline spaces of arbitrary smoothness defined on arbitrary grids. This extends the results recently obtained for spline spaces of maximal smoothness. The presented error estimates are in agreement with the numerical evidence found in the literature that smoother spline spaces exhibit a better approximation behavior per degree of freedom, even for low smoothness of the functions to be approximated. First we introduce results for univariate spline spaces, and then we address multivariate tensor-product spline spaces and isogeometric spline spaces generated by means of a mapped geometry, both in the single-patch and in the multi-patch case.

On stable representations of Bell elements

The paper is concerned with the space of C1 continuous polynomial splines of degree 5 on general triangulations such that the restriction to each triangle corresponds to a Bell element. It is investigated how the standard construction of Bell elements, which is based on prescribing values and derivatives up to order two at the vertices of the triangulation, can be used to define different bases. The most straightforward one is the nodal basis, which lacks stability, but it is shown that this can be improved by scaling the interpolation data. However, the prime focus of the paper is on the construction of B-spline-like bases that depend on the choice of special domain triangles associated with the vertices of the triangulation. A B-spline-like basis consists of functions that are locally supported and form a partition of unity. If the triangulation satisfies certain geometric constraints (e.g. is acute) and the domain triangles are chosen suitably, the basis is also non-negative. The stability analysis of the derived bases is addressed theoretically as well as numerically, the latter with the examples of least square fitting and PDE solving.

Geometrically continuous piecewise Chebyshevian NU(R)BS

By piecewise Chebyshevian splines we mean splines with pieces taken from different Extended Chebyshev spaces all of the same dimension, and with connection matrices at the knots. Within this very large and crucial class of splines, we are more specifically concerned with those which are good for design, in the sense that they possess blossoms, or, equivalently, refinable B-spline bases. In practice, this subclass is known to be characterised by the existence of (infinitely many) piecewise generalised derivatives with respect to which the continuity between consecutive pieces is controlled by identity matrices. Somehow inherent in the previous characterisation, the construction of all associated rational spline spaces creates an equivalence relation between piecewise Chebyshevian spline spaces good for design, among which the famous classical rational splines. We investigate this equivalence relation along with the natural question: Is it or not worthwhile considering the rational framework since it does not enlarge the set of resulting splines? This explains the parentheses inside the acronym NU(R)BS.

Continuous-Domain Signal Reconstruction Using $L_{p}$-Norm Regularization

We focus on the generalized-interpolation problem. There, one reconstructs continuous-domain signals that honor discrete data constraints. This problem is infinite-dimensional and ill-posed. We make it well-posed by imposing that the solution balances data fidelity and some $L_{p}$ -norm regularization. More specifically, we consider $p\geq 1$ and the multi-order derivative regularization operator ${\mathrm{L}}={\mathrm{D}}^{N_{0}}$ . We reformulate the regularized problem exactly as a finite-dimensional one by restricting the search space to a suitable space of polynomial splines with knots on a uniform grid. Our splines are represented in a B-spline basis, which results in a well-conditioned discretization. For a sufficiently fine grid, our search space contains functions that are arbitrarily close to the solution of the underlying problem where our constraint that the solution must live in a spline space would have been lifted. This remarkable property is due to the approximation power of splines. We use the alternating-direction method of multipliers along with a multiresolution strategy to compute our solution. We present numerical results for spatial and Fourier interpolation. Through our experiments, we investigate features induced by the $L_{p}$ -norm regularization, namely, sparsity, regularity, and oscillatory behavior.

Sharp Jackson - Chernykh type inequality for spline approximations on the line

An analog of the Jackson–Chernykh inequality for spline approximations in the space L2($$\mathbb{R}$$) is established in this work. For r ∈ $$\mathbb{N}$$ and σ > 0, we denote by Aσr( f )2 the best approximation of a function f ∈ L2($$\mathbb{R}$$) by the space of splines of degree r of minimal defect with knots $$\frac{{j\pi }}{\sigma }$$, j ∈ $$\mathbb{Z}$$, and by ω( f, δ)2 the first-order modulus of continuity of f in L2($$\mathbb{R}$$). The main result of our work is as follows. For any f ∈ L2($$\mathbb{R}$$), $${{A}_{{\sigma r}}}{{(f)}_{2}}\;\leqslant \;\frac{1}{{\sqrt 2 }}\omega {{\left( {f,\frac{{{{\theta }_{r}}\pi }}{\sigma }} \right)}_{2}},$$ where θr = $$\frac{1}{{\sqrt {1 - \varepsilon _{r}^{2}} }}$$ and er is the positive root of the equation $$\frac{{4{{\varepsilon }^{2}}\left( {\cosh \frac{{\pi \varepsilon }}{\tau } - 1} \right)}}{{\cosh \frac{{\pi \varepsilon }}{\tau } + \cos \frac{\pi }{\tau }}} = \frac{1}{{{{3}^{{2r - 2}}}}},\quad \tau = \sqrt {1 - {{\varepsilon }^{2}}} .$$ The constant $$\frac{1}{{\sqrt 2 }}$$ cannot be reduced on the whole class L2($$\mathbb{R}$$) even by increasing the step of the modulus of continuity.

Optimal subspaces for mean square approximation of classes of differentiable functions on a segment

In this paper, a set of optimal subspaces is specified for L2 approximation of three classes of functions in the Sobolev spaces $$W_{2}^{{(r)}}$$ defined on a segment and subject to certain boundary conditions. A subspace X of a dimension not exceeding n is called optimal for a function class A if the best approximation of A by X is equal to the Kolmogorov n-width of A. These boundary conditions correspond to subspaces of periodically extended functions with symmetry properties. All approximating subspaces are generated by equidistant shifts of a single function. The conditions of optimality are given in terms of Fourier coefficients of a generating function. In particular, we indicate optimal spline spaces of all degrees d $$ \geqslant $$ r – 1 with equidistant knots of several different types.

A Tchebycheffian Extension of Multidegree B-Splines: Algorithmic Computation and Properties

In this paper, we present an efficient and robust approach to compute a normalized B-spline-like basis for spline spaces with pieces drawn from extended Tchebycheff spaces. The extended Tchebycheff spaces and their dimensions are allowed to change from interval to interval. The approach works by constructing a matrix that maps a generalized Bernstein-like basis to the B-spline-like basis of interest. The B-spline-like basis shares many characterizing properties with classical univariate B-splines and may easily be incorporated in existing spline codes. This may contribute to the full exploitation of Tchebycheffian splines in applications, freeing them from the restricted role of an elegant theoretical extension of polynomial splines. Numerical examples are provided that illustrate the procedure described.

A collocation method in spline spaces for the solution of linear fractional dynamical systems

We use a collocation method in refinable spline spaces to solve a linear dynamical system having fractional derivative in time. The method takes advantage of an explicit differentiation rule for the B-spline basis that allows us to efficiently evaluate the collocation matrices appearing in the method. We prove the convergence of the method and show some numerical results.

Splines of Variable Approximation Order and Their Wavelet Decompositions

We construct spline (finite element) spaces of variable approximation order and find necessary and sufficient conditions for pseudosmoothness of such splines. We study embedding of the spline spaces on embedded subdivisions and construct the corresponding wavelet decompositions. The constructions are based on the approximation relations defined on a cell subdivision of a differentiable manifold under the assumption that the multiplicity of the covering by supports of the coordinate functions is variable, which causes the variable approximation order. The spline spaces possess the adaptive approximation property. The notion of pseudosmoothness lead to new families of embedded spaces.