Strongly stable bases for adaptively refined multilevel spline spaces

Abstract

The problem of constructing a normalized hierarchical basis for adaptively refined spline spaces is addressed. Multilevel representations are defined in terms of a hierarchy of basis functions, reflecting different levels of refinement. When the hierarchical model is constructed by considering an underlying sequence of bases Γℓ=0,…,N−1 with properties analogous to classical tensor-product B-splines, we can define a set of locally supported basis functions that form a partition of unity and possess the property of coefficient preservation, i.e., they preserve the coefficients of functions represented with respect to one of the bases Γℓ. Our construction relies on a certain truncation procedure, which eliminates the contributions of functions from finer levels in the hierarchy to coarser level ones. Consequently, the support of the original basis functions defined on coarse grids is possibly reduced according to finer levels in the hierarchy. This truncation mechanism not only decreases the overlapping of basis supports, but it also guarantees strong stability of the construction. In addition to presenting the theory for the general framework, we apply it to hierarchically refined tensor-product spline spaces, under certain reasonable assumptions on the given knot configuration

Adaptive IGAFEM with optimal convergence rates: Hierarchical B-splines

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Local refinement for 3D deformable parametric surfaces

“…This local refinement is not inherent to standard methods as Non-Uniform Rational Basis Splines (NURBS) or classical parameterizations of tensorproduct surfaces. Existing methods to insert points at specific locations were developped in [1,2,3,4,5]. We propose a new generic tensor-product parameterization for surfaces where the degrees of freedom can be locally increased without altering the shape of the surface.…”

Algorithms and Data Structures for Truncated Hierarchical B–splines

“…The present paper focuses on the bivariate tensor-product case. However, the framework can easily be adapted to the multivariate setting and even to more general spline spaces [5,6]. Nevertheless, even if the representation model we are going to introduce may in principle be used to handle non-uniform mesh refinement and even spaces generated by degree elevation, we will consider only the dyadic uniform case throughout this paper.…”

“…Truncated hierarchical B-splines [5,6] form a different basis for the same multilevel B-spline space. The key idea behind this alternative hierarchical construction is to properly exploit the refinable nature of the B-spline basis which allows to express a B-spline of level in terms of (d + 2) 2 functions which belong to level + 1 and of certain binomial coefficients scaled by a factor 2 −d with respect to any dimension.…”

“…They were introduced as a possible extension of normalized tensor-product B-splines to suitably handle the local refinement in adaptive surface approximation algorithms. This multilevel scheme was also generalized and further investigated in [6], where particular attention was devoted to the stability analysis of the proposed hierarchical construction.…”