Spline Spaces Research Articles

Dimensions of biquadratic and bicubic spline spaces over hierarchical T-meshes

This paper discusses the dimensions of biquadratic C1 spline spaces and bicubic C2 spline spaces over hierarchical T-meshes using the smoothing cofactor-conformality method. We obtain the dimension formula of biquadratic C1 spline spaces over hierarchical T-meshes in a concise way. In addition, we provide a dimension formula for bicubic C2 spline spaces over hierarchical T-meshes with fewer restrictions than that in the previous literature. A dimension formula for bicubic C2 spline spaces over a new type of hierarchical T-meshes is also provided.

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.

Planar domain parameterization with THB-splines

Isogeometric analysis uses spline parameterizations to describe the geometry of the physical domain. If such a parameterization is not available, it has to be generated from the domain boundaries. The construction of a good parameterization is crucial since it strongly influences the accuracy of the subsequent analysis. It is of interest to use adaptive techniques such as hierarchical splines for the parameterization, since this facilitates an accurate representation of detailed geometries and potentially improves the efficiency of the subsequent numerical simulation. We use truncated hierarchical (TH) B-splines to generate hierarchical spline spaces, since these functions possess many useful properties, such as linear independence and the fact that they form a non-negative partition of unity. In order to address the trade-off between computational effort and level of difficulty of a specific instance of the problem, we propose three levels of domain parameterization techniques that are based on THB-splines.

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.

Isogeometric mortar methods

The application of mortar methods in the framework of isogeometric analysis is investigated theoretically as well as numerically. For the Lagrange multiplier two choices of uniformly stable spaces are presented, both of them are spline spaces but of a different degree. In one case, we consider an equal order pairing for which a cross point modification based on a local degree reduction is required. In the other case, the degree of the dual space is reduced by two compared to the primal. This pairing is proven to be inf–sup stable without any necessary cross point modification. Several numerical examples confirm the theoretical results and illustrate additional aspects.

Smoothness and error bounds of Martensen splines

Martensen splines Mf of degree n interpolate f and its derivatives up to the order n−1 at a subset of the knots of the spline space, have local support and exactly reproduce both polynomials and splines of degree ≤n. An approximation error estimate has been provided for f∈Cn+1.This paper aims to clarify how well the Martensen splines Mf approximate smooth functions on compact intervals. Assuming that f∈Cn−1, approximation error estimates are provided for Djf,j=0,1,…,n−1, where Dj is the jth derivative operator. Moreover, a set of sufficient conditions on the sequence of meshes are derived for the uniform convergence of DjMf to Djf, for j=0,1,…,n−1.

A HIERARCHICALLY SUPERIMPOSING LOCAL REFINEMENT METHOD FOR ISOGEOMETRIC ANALYSIS

In isogeometric analysis, the tensor-product form of Nonuniform Rational B-spline (NURBS) represents spline surfaces. Due to the nature of the tensor-product, the local refinement in isogeometric analysis has many issues to be resolved. Attempts have been made in this regard, such as T-splines and hierarchical approaches. In this work, a local refinement method for isogeometric analysis based on a superimposing concept is proposed. Local refinements are performed by superimposing hierarchically-created finer overlay meshes onto the regions of high error rather than a change of analysis basis (from NURBS to some other spline space). To employ the superimposing concept as a local refinement strategy in isogeometric analysis, a hierarchical framework to construct overlay meshes is developed, and compatibility conditions across the interfacial boundaries of different levels of meshes is discussed. Through numerical examples, the effectiveness and validity of the proposed method are demonstrated.

An orthogonal basis for non-uniform algebraic-trigonometric spline space

Non-uniform algebraic-trigonometric B-splines shares most of the properties as those of the usual polynomial B-splines. But they are not orthogonal. We construct an orthogonal basis for the n-order(n ≥ 3) algebraic-trigonometric spline space in order to resolve the theoretical problem that there is not an explicit orthogonal basis in the space by now. Motivated by the Legendre polynomials, we present a novel approach to define a set of auxiliary functions, which have simple and explicit expressions. Then the proposed orthogonal splines are given as the derivatives of these auxiliary functions.

On spline-based differential quadrature

In the paper Barrera et al. (2014), a boolean sum differential quadrature method (DQM) was proposed by combining a spline interpolation operator having a fundamental function with minimal compact support and a spline quasi-interpolation operator reproducing the polynomials in the spline space. It is a general framework that provides results that differ from the ones obtained by defining specific schemes with structures which depend on the degree of the B-spline to be considered. The main drawback of these boolean sum DQMs is that the number of evaluation points increases quickly with the degree of the B-spline due to the use of a quasi-interpolation operator. We propose a different construction avoiding this problem and derive explicit results for low degree B-splines.

Image-based goal-oriented adaptive isogeometric analysis with application to the micro-mechanical modeling of trabecular bone

Isogeometric analysis (IGA) of geometrically complex three-dimensional objects is possible when used in combination with the Finite Cell method (FCM). In this contribution we propose a computational methodology to automatically analyze the effective elastic properties of scan-based volumetric objects of arbitrary geometric and topological complexity. The first step is the reconstruction of a smooth geometry from scan-based voxel data using a B-spline level set function. The second step is a goal-oriented adaptive isogeometric linear elastic analysis. Elements are selected for refinement using dual-weighted residual shape function indicators, and hierarchical splines are employed to construct locally refined spline spaces. The proposed methodology is studied in detail for various numerical test cases, including the computation of the effective Young’s modulus of a trabecular bone micro-structure reproduced from μCT-scan data.

Wavelet Decomposition of Hermite-Type Splines

The wavelet decompositions of spaces of Hermite-type splines of class C n (α,β), that are defined by a (2n+2)-component vector valued function φ(t) ∈ C n (α,β) by means of a grid X (not necessarily uniform) on (α,β) are considered. The basic wavelets obtained are compactly supported. The decomposition and reconstruction formulas are given.

Bounds on the Dimension of Trivariate Spline Spaces: A Homological Approach

We consider the vector space of globally differentiable piecewise polynomial functions defined on a three-dimensional polyhedral domain partitioned into tetrahedra. We prove new lower and upper bounds on the dimension of this space by applying homological techniques. We give an insight of different ways of approaching this problem by exploring its connections with the Hilbert series of ideals generated by powers of linear forms, fat points, the so-called Froberg–Iarrobino conjecture, and the weak Lefschetz property.

TDHB-splines: The truncated decoupled basis of hierarchical tensor-product splines

We introduce a novel basis for multivariate hierarchical tensor-product spline spaces. Our construction combines the truncation mechanism (Giannelli et al., 2012) with the idea of decoupling basis functions (Mokriš et al., 2014). While the first mechanism ensures the partition of unity property, which is essential for geometric modeling applications, the idea of decoupling allows us to obtain a richer set of basis functions than previous approaches. Consequently, we can guarantee the completeness property of the novel basis for large classes of multi-level spline spaces. In particular, completeness is obtained for the multi-level spline spaces defined on T-meshes for hierarchical splines of (multi-)degree p for example (i) with single knots and p-adic refinement and (ii) with knots of multiplicity m≥(p+1)/3 and dyadic refinement (where each cell to be refined is subdivided into 2d cells, with d being the number of variables) without any further restriction on the mesh configuration. Both classes (i), (ii) include multivariate quadratic hierarchical tensor-splines with dyadic refinement.

Adaptively refined multilevel spline spaces from generating systems

The truncated basis of adaptively refined multilevel spline spaces was introduced by Giannelli et al. (2012, 2014). It possesses a number of advantages, including the partition of unity property, decreased support of the basis functions, preservation of coefficients and strong stability, that may make it highly useful for geometric modeling and numerical simulation. We generalize this construction to hierarchies of spaces that are spanned by generating systems that potentially possess linear dependencies. This generalization requires a modified framework, since the existing approach relied on the linear independence of the functions generating the spaces in the hierarchy. Many results, such as the preservation of coefficients, can be extended to the more general setting. As applications of the modified framework, we introduce a hierarchy of hierarchical B-splines, which enables us to perform local refinement in the presence of features, and we also extend the adaptive multilevel framework to spaces spanned by Zwart–Powell (ZP) elements, which are special box splines defined on the criss-cross grid. In the latter case we show how to identify the linear dependencies that are present in the truncated hierarchical generating system and use this result to perform adaptive surface fitting with multilevel ZP elements.

Approximation with diversified B-splines

When approximating functions defined on some domain Ω⊂Rd, standard tensor product splines reveal sub-optimal behavior, in particular, if Ω is non-convex. As an alternative, we suggest a natural diversification strategy for the B-spline basis {Bi}i. It is grounded on employing a separate copy Bi,γ of Bi for every connected component γ of its support suppBi∩Ω. In the bivariate case, which is important for applications, this process enhances the spline space to a crucial extent. Concretely, we prove that the error in uniform tensor product spline approximation of a function f:R2⊃Ω→R can be bounded in terms of the pure partial derivatives of f, where the constant depends neither on the shape of Ω nor on the knot grid. An example shows that a similar result cannot hold true for higher dimensions, even if the domain is convex and has a smooth boundary.

A Hermite interpolatory subdivision scheme for [formula omitted]-quintics on the Powell–Sabin 12-split

In order to construct a C1-quadratic spline over an arbitrary triangulation, one can split each triangle into 12 subtriangles, resulting in a finer triangulation known as the Powell–Sabin 12-split. It has been shown previously that the corresponding spline surface can be plotted quickly by means of a Hermite subdivision scheme (Dyn and Lyche, 1998). In this paper we introduce a nodal macro-element on the 12-split for the space of quintic splines that are locally C3 and globally C2. For quickly evaluating any such spline, a Hermite subdivision scheme is derived, implemented, and tested in the computer algebra system Sage. Using the available first derivatives for Phong shading, visually appealing plots can be generated after just a couple of refinements.

On the completeness of hierarchical tensor-product [formula omitted]-splines

Given a grid in Rd, consisting of d bi-infinite sequences of hyperplanes (possibly with multiplicities) orthogonal to the d axes of the coordinate system, we consider the spaces of tensor-product spline functions of a given degree on a multi-cell domain. Such a domain consists of finite set of cells which are defined by the grid. A piecewise polynomial function belongs to the spline space if its polynomial pieces on adjacent cells have a contact according to the multiplicity of the hyperplanes in the grid. We prove that the connected components of the associated set of tensor-product B-splines, whose support intersects the multi-cell domain, form a basis of this spline space. More precisely, if the intersection of the support of a tensor-product B-spline with the multi-cell domain consists of several connected components, then each of these components contributes one basis function. In order to establish the connection to earlier results, we also present further details relating to the three-dimensional case with single knots only.A hierarchical B-spline basis is defined by specifying nested hierarchies of spline spaces and multi-cell domains. We adapt the techniques from Giannelli and Jüttler (2013) to the more general setting and prove the completeness of this basis (in the sense that its span contains all piecewise polynomial functions on the hierarchical grid with the smoothness specified by the grid and the degrees) under certain assumptions on the domain hierarchy.Finally, we introduce a decoupled version of the hierarchical spline basis that allows to relax the assumptions on the domain hierarchy. In certain situations, such as quadratic tensor-product splines, the decoupled basis provides the completeness property for any choice of the domain hierarchy.

Hierarchical B-splines on regular triangular partitions

Multivariate splines have a wide range of applications in function approximation, finite element analysis and geometric modeling. They have been extensively studied in the last several decades, and specially the theory on bivariate B-splines over regular triangular partition is well developed. However, the above mentioned splines do not have local refinement property – a property that is very important in adaptive function approximation and level of detailed representation of geometric models. In this paper, we introduce the concept of hierarchial bivariate splines over regular triangular partitions and construct basis functions of such spline space that satisfy some nice properties. We provide some examples of hierarchical splines over triangular partitions in surface fitting and in solving numerical PDEs, and the results turn out to be promising.

Spline spaces with mixed orders of continuity over T-meshes

In this paper, we introduce the concept of spline spaces with mixed orders of continuity over T-meshes. Then, the dimensions of the cubic spline spaces with continuity of order one and locally discontinuous over hierarchical T-meshes are presented by the B-net method. From the viewpoint of processing geometry data, a non-negative basis set with local support and partition of unity is constructed. Finally, the behavior of this type of spline is analyzed with the help of examples in image processing and finite element analysis.

A wavelet renement algorithms of the rst order spline spaces

The goal of this paper is to obtain a wavelet decomposition (wavelet renement) of the chain of embedded spaces of splines for an arbitrary renement of a nonuniform grid, and to derive the corresponding decomposition and reconstruction formulas, to construct wavelet decompositions and decomposition and reconstruction algorithms in the case of an innite ow for a grid on an open interval and a nite ow for a grid on a segment.