Linear dependence of bivariate Minimal Support and Locally Refined B-splines over LR-meshes

Abstract

The focus on locally refined spline spaces has grown rapidly in recent years due to the need in Isogeoemtric analysis (IgA) of spline spaces with local adaptivity: a property not offered by the strict regular structure of tensor product B-spline spaces. However, this flexibility sometimes results in collections of B-splines spanning the space that are not linearly independent. In this paper we address the minimal number of B-splines that can form a linear dependence relation for Minimal Support B-splines (MS B-splines) and for Locally Refined B-splines (LR B-splines) on LR-meshes. We show that the minimal number is six for MS Bsplines and eight for LR B-splines. Further results are established to help detecting collections of B-splines that are linearly independent.

Adaptive refinement with locally linearly independent LR B-splines: Theory and applications

“…In particular, the region corresponds to the support of an LR B-spline that has many nested LR B-splines in it. One can prove the existence of the linear dependence relation by computing the spline space dimension and the number of LR B-splines defined on the mesh as explained in the examples of [20]. This configuration can be reproduced for any bidegree (p 1 , p 2 ) with p k ≥ 4 for k = 1, 2.…”

“…In this section, we introduce locally refined B-splines, or in short LR B-splines, and discuss several of their properties, following the terminology from [20]. We denote by Π p the space of univariate polynomials of degree less than or equal to p, and by Π p p p the space of bivariate polynomials of degrees less than or equal to p p p = (p 1 , p 2 ) component-wise.…”

“…In [10] an efficient algorithm to seek and destroy linear dependence relations has been introduced, but it does not handle every possible locally refined mesh. In [20] a first analysis on the necessary conditions for encountering a linear dependence relation has been presented. There, it has also been proved that, for any bidegree, a linear dependence relation in the LR B-spline set involves at least 8 functions.…”

Space Mapping of Spline Spaces over Hierarchical T-meshes

“…In practice, linear dependence of LR B-splines can be controlled and much knowledge exists with respect to mesh configurations resulting in linear dependent LR B-splines. In [16], a first analysis on the necessary conditions for encountering a linear dependence relation has been presented. In [15], different properties of the LR-splines are analyzed: in particular the coefficients for polynomial representations and their relation with other properties such as linear independence and the number of B-splines covering each element.…”

Adaptive refinement with locally linearly independent LR B-splines: Theory and applications

“…By allowing local insertions in the underlying mesh, the approximation efficiency is dramatically improved as one avoids the wasting of degrees of freedom by increasing the number of basis functions only where rapid and large variations occur in the analyzed object. Nevertheless, the adoption of LR B-splines for simulation purposes in the Isogeometric Analysis (IgA) framework [15] is hindered by the risk of linear dependence relations [20]. Although a complete characterization of linear independence is still not available, the local linear independence of the basis functions is guaranteed when the underlying Locally Refined (LR) mesh has the so-called Non-Nested-Support (N 2 S) property [2,3].…”