Local Hierarchical h-Refinements in IgA Based on Generalized B-Splines

Abstract

In this paper we construct multilevel representations in terms of a hierarchy of tensor-product generalized B-splines. These representations combine the positive properties of a non-rational model with the possibility of dealing with local refinements. We discuss their use in the context of isogeometric analysis.

Generalized spline spaces over T-meshes: Dimension formula and locally refined generalized B-splines

“…In particular, it has already been shown in the literature that the construction and properties of (truncated) hierarchical B-splines and analysis-suitable T-splines also extend to the generalized spline context, see [15,26,35] and [3,4], respectively.…”

“…In particular, GB-splines support (locally refined) hierarchical structures in the same way as polynomial B-splines, see [26] (and also [15,35]). T-spline structures based on GB-splines have been addressed in [3,4].…”

“…Thanks to the above properties, tensor-products of generalized B-splines are also an interesting problem-dependent alternative to tensor-product polynomial B-splines and NURBS in isogeometric analysis (IgA), see [9,[24][25][26]. Introduced nearly a decade ago in a seminal paper by Hughes et al [16], IgA is nowadays a well-established paradigm for the analysis of problems governed by partial differential equations (PDEs), see, e.g., [10] and references therein.…”

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.…”

Reissner–Mindlin shell implementation and energy conserving isogeometric multi‐patch coupling

“…The basics of local h-refinement with T-splines was first demonstrated in [19], and polynomial splines over hierarchical T-meshes are used in [20] for local refinements. An algorithm for T-spline refinement that solves problems with excessive propagation of control points is presented in [21].Instead of using T-splines, IGA mesh adaptivity is also investigated by using hierarchical adaptive local refinement with generalized B-splines [22] and with the extension of hierarchical B-splines [23][24][25]. Hierarchical adaptive refinement is also examined in [26] with locally refined B-splines and approaches based on localized multigrids.…”