Spline Spaces Research Articles

Orthogonal projectors onto spaces of periodic splines

The main result of this paper is a proof that for any integrable function f on the torus, any sequence of its orthogonal projections (P˜nf) onto periodic spline spaces with arbitrary knots Δ˜n and arbitrary polynomial degree converges to f almost everywhere with respect to the Lebesgue measure, provided the mesh diameter ∣Δ˜n∣ tends to zero. We also give a new and simpler proof of the fact that the operators P˜n are bounded on L∞ independently of the knots Δ˜n.

Semialgebraic splines

We study bivariate splines over partitions defined by arcs of irreducible algebraic curves, which we call semialgebraic splines. Such splines were first considered by Wang, Chui, and Stiller. We compute the dimension of the space of semialgebraic splines in two extreme cases when the cell decomposition has a single interior vertex. If the forms defining the edges span a two-dimensional space of forms of degree n, then we compute the dimension of the spline space in every degree. In the other extreme, the curves have distinct slopes at the central vertex and do not simultaneously vanish at any other point. In this case we give a formula for the dimension of the spline space in large degree and bound how large the degree must be for the formula to be correct. We also study the dimension of the spline space in the case of a single interior vertex in some examples where the curves do not satisfy either extreme. The results are derived using commutative and homological algebra.

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.

Gaussian quadrature rules for [formula omitted] quintic splines with uniform knot vectors

We provide explicit quadrature rules for spaces of C1 quintic splines with uniform knot sequences over finite domains. The quadrature nodes and weights are derived via an explicit recursion that avoids numerical solvers. Each rule is optimal, that is, requires the minimal number of nodes, for a given function space. For each of n subintervals, generically, only two nodes are required which reduces the evaluation cost by 2/3 when compared to the classical Gaussian quadrature for polynomials over each knot span. Numerical experiments show fast convergence, as n grows, to the “two-third” quadrature rule of Hughes et al. (2010) for infinite domains.

Adaptive scattered data fitting by extension of local approximations to hierarchical splines

We introduce an adaptive scattered data fitting scheme as extension of local least squares approximations to hierarchical spline spaces. To efficiently deal with non-trivial data configurations, the local solutions are described in terms of (variable degree) polynomial approximations according not only to the number of data points locally available, but also to the smallest singular value of the local collocation matrices. These local approximations are subsequently combined without the need of additional computations with the construction of hierarchical quasi-interpolants described in terms of truncated hierarchical B-splines. A selection of numerical experiments shows the effectivity of our approach for the approximation of real scattered data sets describing different terrain configurations.

Hermite Type Spline Spaces over Rectangular Meshes with Complex Topological Structures

AbstractMotivated by the magneto hydrodynamic (MHD) simulation for Tokamaks with Isogeometric analysis, we present splines defined over a rectangular mesh with a complex topological structure, i.e., with extraordinary vertices. These splines are piecewise polynomial functions of bi-degree (d,d) andparameter continuity. And we compute their dimension and exhibit basis functions called Hermite bases for bicubic spline spaces. We investigate their potential applications for solving partial differential equations (PDEs) over a physical domain in the framework of Isogeometric analysis. For instance, we analyze the property of approximation of these spline spaces for theL2-norm; we show that the optimal approximation order and numerical convergence rates are reached by setting a proper parameterization, although the fact that the basis functions are singular at extraordinary vertices.

Subdivision and Spline Spaces

A standard construction in approximation theory is mesh refinement. For a simplicial or polyhedral mesh $$\varDelta \subseteq {\mathbb R}^k$$ , we study the subdivision $$\varDelta '$$ obtained by subdividing a maximal cell of $$\varDelta $$ . We give sufficient conditions for the module of splines on $$\varDelta '$$ to split as the direct sum of splines on $$\varDelta $$ and splines on the subdivided cell. As a consequence, we obtain dimension formulas and explicit bases for several commonly used subdivisions and their multivariate generalizations.

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.

Robust Multigrid for Isogeometric Analysis Based on Stable Splittings of Spline Spaces

We present a robust and efficient multigrid method for single-patch isogeometric discretizations using tensor product B-splines of maximum smoothness. Our method is based on a stable splitting of the spline space into a large subspace of “interior” splines which satisfy a robust inverse inequality, as well as one or several smaller subspaces which capture the boundary effects responsible for the spectral outliers which occur in isogeometric analysis. We then construct a multigrid smoother based on an additive subspace correction approach, applying a different smoother to each of the subspaces. For the interior splines, we use a mass smoother, whereas the remaining components are treated with suitably chosen Kronecker product smoothers or direct solvers. We prove that the resulting multigrid method exhibits iteration numbers which are robust with respect to the spline degree and the mesh size. Furthermore, it can be efficiently realized for discretizations of problems in arbitrarily high geometric dimension. Some numerical examples illustrate the theoretical results and show that the iteration numbers also scale relatively mildly with the problem dimension.

Optimal spline spaces of higher degree for [formula omitted] [formula omitted]-widths

In this paper we derive optimal subspaces for Kolmogorov n-widths in the L2 norm with respect to sets of functions defined by kernels. This enables us to prove the existence of optimal spline subspaces of arbitrarily high degree for certain classes of functions in Sobolev spaces of importance in finite element methods. We construct these spline spaces explicitly in special cases.

Anisotropic spline approximation with non-uniform B-splines

Recently the author and U. Reif introduced the concept of diversification of uniform tensor product B-splines. Based on this concept, we give a new constructive modification of non-uniform B-splines. The resulting spline spaces are perfectly fitted for the approximation of functions defined on domains . We build a bounded quasi-interpolant and prove that for our spline spaces an anisotropic error estimate in the -norm, , is valid. In particular, we show that the constant of the error estimate does not depend on the shape of or the knot grid.

Optimal and reduced quadrature rules for tensor product and hierarchically refined splines in isogeometric analysis

We continue the study initiated in Hughes et al. (2010) in search of optimal quadrature rules for tensor product and hierarchically refined splines in isogeometric analysis. These rules are optimal in the sense that there exists no other quadrature rule that can exactly integrate the elements of the given spline space with fewer quadrature points. We extend the algorithm presented in Hughes et al. (2010) with an improved starting guess, which combined with arbitrary precision arithmetic, results in the practical computation of quadrature rules for univariate non-uniform splines up to any precision. Explicit constructions are provided in sixteen digits of accuracy for some of the most commonly used uniform spline spaces defined by open knot vectors. We study the efficacy of the proposed rules in the context of full and reduced quadrature applied to two- and three-dimensional diffusion–reaction problems using tensor product and hierarchically refined splines, and prove a theorem rigorously establishing the stability and accuracy of the reduced rules.

Hierarchical spline spaces: quasi-interpolants and local approximation estimates

A local approximation study is presented for hierarchical spline spaces. Such spaces are composed of a hierarchy of nested spaces and provide a flexible framework for local refinement in any dimensionality. We provide approximation estimates for general hierarchical quasi-interpolants expressed in terms of the truncated hierarchical basis. Under some mild assumptions, we prove that such hierarchical quasi-interpolants and their derivatives possess optimal local approximation power in the general q-norm with 1≤q≤ź$1\leq q\leq \infty $. In addition, we detail a specific family of hierarchical quasi-interpolants defined on uniform hierarchical meshes in any dimensionality. The construction is based on cardinal B-splines of degree p and central factorial numbers of the first kind. It guarantees polynomial reproduction of degree p and it requires only function evaluations at grid points (odd p) or half-grid points (even p). This results in good approximation properties at a very low cost, and is illustrated with some numerical experiments.

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.

Adaptive isogeometric analysis with hierarchical box splines

Isogeometric analysis is a recently developed framework based on finite element analysis, where the simple building blocks in geometry and solution space are replaced by more complex and geometrically-oriented compounds. Box splines are an established tool to model complex geometry, and form an intermediate approach between classical tensor-product B-splines and splines over triangulations. Local refinement can be achieved by considering hierarchically nested sequences of box spline spaces. Since box splines do not offer special elements to impose boundary conditions for the numerical solution of partial differential equations (PDEs), we discuss a weak treatment of such boundary conditions. Along the domain boundary, an appropriate domain strip is introduced to enforce the boundary conditions in a weak sense. The thickness of the strip is adaptively defined in order to avoid unnecessary computations. Numerical examples show the optimal convergence rate of box splines and their hierarchical variants for the solution of PDEs.

An isogeometric method for linear nearly-incompressible elasticity with local stress projection

In this paper, we propose an isogeometric method for solving the linear nearly-incompressible elasticity problem. The method is similar to the B̄ formulation where the volumetric strain is projected on a lower degree spline space in order to prevent volumetric locking. In our method, we adopt a local projection on a coarser mesh, chosen in order to guarantee optimal convergence. Moreover the locality of the projector allows to maintain the sparsity of the stiffness matrix, that is, the efficiency of the method. The analysis of the method is based on the inf–sup stability of the associated mixed formulation via a macro-element technique for spline functions. The numerical tests confirm the theory of the method.

Fast formation of isogeometric Galerkin matrices by weighted quadrature

In this paper we propose an algorithm for the formation of matrices of isogeometric Galerkin methods. The algorithm is based on three ideas. The first is that we perform the external loop over the rows of the matrix. The second is that we calculate the row entries by weighted quadrature. The third is that we exploit the (local) tensor product structure of the basis functions. While all ingredients have a fundamental role for computational efficiency, the major conceptual change of paradigm with respect to the standard implementation is the idea of using weighted quadrature: the test function is incorporated in the integration weight while the trial function, the geometry parametrization and the PDEs coefficients form the integrand function. This approach is very effective in reducing the computational cost, while maintaining the optimal order of approximation of the method. Analysis of the cost is confirmed by numerical testing, where we show that, for p large enough, the time required by the floating point operations is less than the time spent in unavoidable memory operations (the sparse matrix allocation and memory write). The proposed algorithm allows significant time saving when assembling isogeometric Galerkin matrices for all the degrees of the test spline space and paves the way for a use of high-degree k-refinement in isogeometric analysis.

Splines over regular triangulations in numerical simulation

We investigate the use of smooth spline spaces over regular triangulations as a tool in (isogeometric) Galerkin methods. In particular, we focus on box splines over three-directional meshes. Box splines are multivariate generalizations of univariate cardinal B-splines sharing the same properties. Tensor-product B-splines with uniform knots are a special case of box splines. The use of box splines over three-directional meshes has several advantages compared with tensor-product B-splines, including enhanced flexibility in the treatment of the geometry and stiffness matrices with stronger sparsity. Boundary conditions are imposed in a weak form to avoid the construction of special boundary functions. We illustrate the effectiveness of the approach by means of a selection of numerical examples.

Refinable spaces and local approximation estimates for hierarchical splines

We study the local approximation properties in hierarchical spline spaces through multiscale quasi-interpolation operators. This construction suggests the analysis of a subspace of the classical hierarchical spline space (Vuong et al. (2011) A hierarchical approach to adaptive local refinement in isogeometric analysis. Comput. Methods Appl. Mech. Eng., 200, 3554–3567) which still satisfies the essential properties of the full space. The B-spline basis of such a subspace can be constructed using parent–children relations only, making it well adapted to local refinement algorithms.

Discretized Lavrent’ev regularization for the autoconvolution equation

Lavrent’ev regularization for the autoconvolution equation was considered by Janno J. in Lavrent’ev regularization of ill-posed problems containing nonlinear near-to-monotone operators with application to autoconvolution equation, Inverse Prob. 2000;16:333–348. Here this study is extended by considering discretization of the Lavrent’ev scheme by splines. It is shown how to maintain the known convergence rate by an appropriate choice of spline spaces and a proper choice of the discretization level. For piece-wise constant splines the discretized equation allows for an explicit solver, in contrast to using higher order splines. This is used to design a fast implementation by means of post-smoothing, which provides results, which are indistinguishable from results obtained by direct discretization using cubic splines.