Bivariate Spline Spaces Research Articles

Cell reducing and the dimension of the C1 bivariate spline space

In this paper, the problem of determining the dimension of the space Sn1(△), n ≥ 3 of bivariate C1 splines of degree ≤ n over a triangulation △ is considered. The piecewise polynomials are represented as blossoms, and the smoothness conditions are written as a system of linear equations. The rank of the system matrix is analysed by repeatedly reducing small subtriangulations (cells) at the boundary of a triangulation. It is shown that the dimension of the bivariate spline space Sn1(△), n ≥ 3 is equal to Schumaker’s lower bound for a large class of triangulations.

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.

Bivariate Spline Finite Element Solver for Linear Hyperbolic Equations in Two-Dimensional Spaces

In this paper, a finite element method using bivariate spline on domains is proposed for solving linear hyperbolic equations in two-dimensional spaces. Bivariate spline space $$S_{4}^{2,3} (\Delta_{mn}^{(2)} )$$ is constructed. It not only satisfies homogeneous boundary constraints but also satisfies interpolating boundary conditions on type-2 triangulations. Two examples are shown to confirm the correctness of theoretical conclusions. This means that spline method is efficient and feasible to solve 2D linear hyperbolic equations.

The bivariate quadraticC1spline spaces with stable dimensions on the triangulations

The dimension is a basic problem in the theory of the bivariate spline space. In general, the dimension of the bivariate spline space defined on the triangulation is unstable or with singularity. In this paper, we consider the dimensions of the bivariate quadratic C 1 spline spaces defined on the triangulations. Our main result is when the degree of each interior vertex of the non-degenerate triangulation is at least 6, the dimension of the corresponding bivariate quadratic C 1 spline space is stable and equal to the number of the boundary vertices plus 3. We also give an example to show that the non-degenerate condition is necessary.

A numerical method for solving the time variable fractional order mobile–immobile advection–dispersion model

In this article, we proposed a new numerical method to obtain the approximation solution for the time variable fractional order mobile–immobile advection–dispersion model based on reproducing kernel theory and collocation method. The equation is obtained from the standard advection–dispersion equation (ADE) by adding the Coimbra's variable fractional derivative in time of order γ(x,t)∈[0,1]. In order to solve this kind of equation, we discuss and derive the ε-approximate solution in the form of series with easily computable terms in the bivariate spline space. At the same time, the stability and convergence of the approximation are investigated. Finally, numerical examples are provided to show the accuracy and effectiveness.

Quasi-Interpolation Operators for Bivariate Quintic Spline Spaces and Their Applications

Splines and quasi-interpolation operators are important both in approximation theory and applications. In this paper, we construct a family of quasi-interpolation operators for the bivariate quintic spline spaces S 5 3 ( Δ m n ( 2 ) ) . Moreover, the properties of the proposed quasi-interpolation operators are studied, as well as its applications for solving the two-dimensional Burgers’ equation and image reconstruction. Some numerical examples show that these methods, which are easy to implement, provide accurate results.

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.

Characterization of bivariate hierarchical quartic box splines on a three-directional grid

We consider the adaptive refinement of bivariate quartic C2-smooth box spline spaces on the three-directional (type-I) grid G. The polynomial segments of these box splines belong to a certain subspace of the space of quartic polynomials, which will be called the space of special quartics. Given a bounded domain Ω⊂R2 and finite sequence (Gℓ)ℓ=0,…,N of dyadically refined grids, we obtain a hierarchical grid by selecting mutually disjoint cells from all levels such that their union covers the entire domain. Using a suitable selection procedure allows to define a basis spanning the hierarchical box spline space. The paper derives a characterization of this space. Under certain mild assumptions on the hierarchical grid, the hierarchical spline space is shown to contain all C2-smooth functions whose restrictions to the cells of the hierarchical grid are special quartic polynomials. Thus, in this case we can give an affirmative answer to the completeness questions for the hierarchical box spline basis.

Formula omitted] cubic splines on Powell–Sabin triangulations

A bivariate C1 cubic spline space on a triangulation with Powell–Sabin refinement which extends the well-known C1 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.

Linear differential operators on bivariate spline spaces and spline vector fields

We consider the application of standard differentiation operators to spline spaces and spline vector fields defined on triangulations in the plane. In particular, we explore the use of Bernstein–Bezier techniques for answering questions such as: What are the images or the kernels, and their dimensions, of partial derivative, gradient, divergence, curl, or Laplace, operators. We also describe a particular continuous piecewise quadratic finite element whose nodal parameters are function and divergence (or curl) values.

Bases of T-meshes and the refinement of hierarchical B-splines

In this paper we consider spaces of bivariate splines of bi-degree (m,n) with maximal order of smoothness over domains associated to a two-dimensional grid. We define admissible classes of domains for which suitable combinatorial technique allows us to obtain the dimension of such spline spaces and the number of tensor-product B-splines acting effectively on these domains. Following the strategy introduced recently by Giannelli and Jüttler, these results enable us to prove that under certain assumptions about the configuration of a hierarchical T-mesh the hierarchical B-splines form a basis of bivariate splines of bi-degree (m,n) with maximal order of smoothness over this hierarchical T-mesh. In addition, we derive a sufficient condition about the configuration of a hierarchical T-mesh that ensures a weighted partition of unity property for hierarchical B-splines with only positive weights.

Scattered data interpolation by bivariate splines with higher approximation order

Given a set of scattered data, we usually use a minimal energy method to find a Lagrange interpolation in a bivariate spline space over a triangulation of the scattered data locations. It is known that the approximation order of the minimal energy spline interpolation is only 2 in terms of the size of triangulation. To improve this order of approximation, we propose several new schemes in this paper. Mainly we follow the ideas of clamped cubic interpolatory splines and not-a-knot interpolatory splines in the univariate setting and extend them to the bivariate setting. In addition, instead of the energy functional of the second order, we propose to use higher order versions. We shall present some theoretical analysis as well as many numerical results to demonstrate that our new interpolation schemes indeed have a higher order of approximation than the classic minimal energy interpolatory spline.

Bases and dimensions of bivariate hierarchical tensor-product splines

We prove that the dimension of bivariate tensor-product spline spaces of bi-degree (d,d) with maximum order of smoothness on a multi-cell domain (more precisely, on a set of cells from a tensor-product grid) is equal to the number of tensor-product B-spline basis functions, defined by only single knots in both directions, acting on the considered domain. A certain reasonable assumption on the configuration of the cells is required.This result is then generalized to the case of piecewise polynomial spaces, with the same smoothness properties mentioned above, defined on a multi-grid multi-cell domain (more precisely, on a set of cells from a hierarchy of tensor-product grids). Again, a certain reasonable assumption regarding the configuration of cells is needed.Finally, it is observed that this construction corresponds to the classical definition of hierarchical B-spline bases. This allows to conclude that this basis spans the full space of spline functions on multi-grid multi-cell domains under reasonable assumptions.

B-spline bases for unequally smooth quadratic spline spaces on non-uniform criss-cross triangulations

In this paper, we investigate bivariate quadratic spline spaces on non-uniform criss-cross triangulations of a bounded domain with unequal smoothness across inner grid lines. We provide the dimension of the above spaces and we construct their local bases. Moreover, we propose a computational procedure to get such bases. Finally we introduce spline spaces with unequal smoothness also across oblique mesh segments.

On the problem of instability in the dimension of a spline space over a T-mesh

In this paper, we discuss the problem of instability in the dimension of a spline space over a T-mesh. For bivariate spline spaces S(5,5,3,3) and S(4,4,2,2), the instability in the dimension is shown over certain types of T-meshes. This result could be considered as an attempt to answer the question of how large the polynomial degree (m,m′) should be relative to the smoothness (r,r′) to make the dimension of a spline space stable. We show in particular that the bound m≥2r+1 and m′≥2r′+1 are optimal.

An Upper Bound of the Bezout Number for Piecewise Algebraic Curves over a Rectangular Partition

A piecewise algebraic curve is a curve defined by the zero set of a bivariate spline function. Given two bivariate spline spaces (Δ) over a domainDwith a partition Δ, the Bezout number BN(m,r;n,t;Δ) is defined as the maximum finite number of the common intersection points of two arbitrary piecewise algebraic curves (Δ). In this paper, an upper bound of the Bezout number for piecewise algebraic curves over a rectangular partition is obtained.

CENTROID TRIANGULATIONS FROM k-SETS

Given a set V of n points in the plane, no three of them being collinear, a convex inclusion chain of V is an ordering of the points of V such that no point belongs to the convex hull of the points preceding it in the ordering. We call k-set of the convex inclusion chain, every k-set of an initial subsequence of at least k points of the ordering. We show that the number of such k-sets (without multiplicity) is an invariant of V, that is, it does not depend on the choice of the convex inclusion chain. Moreover, this number is equal to the number of regions of the order-k Voronoi diagram of V (when no four points are cocircular). The dual of the order-k Voronoi diagram belongs to the set of so-called centroid triangulations that have been originally introduced to generate bivariate simplex spline spaces. We show that the centroids of the k-sets of a convex inclusion chain are the vertices of such a centroid triangulation. This leads to the currently most efficient algorithm to construct particular centroid triangulations of any given point set; it runs in O(n log n + k(n - k) log k) worst case time.

On the dimensions of bivariate spline spaces and the stability of the dimensions

In this paper, the dimensions of bivariate spline spaces are studied using the Smoothing Cofactor-Conformality method. Based on the analysis on the conformality condition at one interior vertex, the stability (or singularity to the contrary) of the dimensions of general spline spaces is discussed in detail. By the aid of directed partition some new results on dimensions are obtained with the corresponding constraints depending on the degree, the smoothness order of the spline spaces and the structure of the partition as well.

On the Dimension of Bivariate <i>C</i><sup>1</sup> Cubic Spline Space with Homogeneous Boundary Conditions over a CT Triangulation

A bivariate spline is a piecewise polynomial with some smoothness de ned on a parti- tion. In this paper, we mainly study the dimensions of bivariate C1 cubic spline spaces S1;0 3 (CT ) and S1;1 3 (CT ) with homogeneous boundary conditions over CT by using interpolating technique, where CT stands for a CT triangulation. The dimensions are related with the numbers of the inter vertices and the singular boundary vertices. The results of this paper can be applied in many elds such as the nite element method for partial di erential equation, computer aided design, numerical approximation, and so on.

On the Dimension of Bivariate Weak Spline Space Over Regular Rectilinear Partition

In this paper, we mainly study the dimensions of bivariate weak spline spaces \({W_k^\mu(I_{1}\Delta)}\) (k ≥ 2μ+1) and \({W_{2}^{1} (I_{1}^{*}\Delta)}\) by using the smoothing cofactor-conformality method, where I1Δ and \({I_{1}^{*} \Delta}\) are regular rectilinear partitions with appointed point sets. Some future works relative to bivariate weak splines are also listed at the end of this paper.