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 Juettler, 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

Bivariate hierarchical Hermite spline quasi--interpolation

Spline quasi-interpolation (QI) is a general and powerful approach for the construction of low cost and accurate approximations of a given function. In order to provide an efficient adaptive approximation scheme in the bivariate setting, we consider quasi-interpolation in hierarchical spline spaces. In particular, we study and experiment the features of the hierarchical extension of the tensor-product formulation of the Hermite BS quasi-interpolation scheme. The convergence properties of this hierarchical operator, suitably defined in terms of truncated hierarchical B-spline bases, are analyzed. A selection of numerical examples is presented to compare the performances of the hierarchical and tensor-product versions of the scheme

Inf-sup stability of isogeometric Taylor-Hood and Sub-Grid methods for the Stokes problem with hierarchical splines

In this article, we prove the inf-sup stability of an adaptive isogeometric discretization of the Stokes problem. The discretization is based on the hierarchical generalization of the isogeometric Taylor-Hood and Sub-Grid elements, which were described by Bressan & Sangalli (2013, Isogeometric discretizations of the Stokes problem: stability analysis by the macroelement technique. IMA J. Numer. Anal., 33, 629- 651) for tensor-product splines. In order to extend the existing proof to the hierarchical setting, we need to adapt some of the steps considerably. In particular, the required local approximation estimate is obtained by analysing the properties of the quasi-interpolant of Speleers & Manni (2016, Effortless quasi-interpolation in hierarchical spaces. Numer. Math., 132, 155-184) with respect to certain Sobolev norms. In addition to the theoretical results, we also perform numerical tests in order to analyse the dependency of the inf-sup constant on the mesh regularity assumptions. Finally, the article also presents a numerical convergence test of the resulting adaptive method on a T-shaped domain

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-mesh with fewer restrictions than that in the previous literature. A dimension formula for bicubic C2 spline spaces over a new type hierarchical T-mesh is also provided.Comment: 21 pages, 19 figure

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

Representing Polynomial Splines over Infinite Hierarchical Meshes by Finite Automata

We propose a data structure based on finite automata for representing polynomial splines over infinite hierarchical meshes. It allows to store and operate such splines using only finite amount of memory. This naturally extends a classical framework of hierarchical tensor product B-splines for infinite meshes in a way suitable for computing