1. Introduction
T-splines (Sederberg et al., 2003) have been introduced as a free-form geometric technology and are one of the most promising features in the Isogeometric Analysis (IGA) framework introduced by Hughes et al. (2005), Cottrell et al. (2009). At present, the main interest in IGA is in finding discrete function spaces that integrate well into CAD applications and, at the same time, can be used for Finite Element Analysis. Throughout the last years, hierarchical B-Splines (Scott et al., 2014; Kuru et al., 2014) and LR-Splines (Dokken et al., 2013; Johannessen et al., 2014) have arisen as alternative approaches to T-Splines for the establishment of an adaptive B-Spline technology. While none of these strategies has outperformed the other competing approaches until today, this paper aims to push forward and motivate the T-Spline technology. Since T-splines can be locally refined (Sederberg et al., 2004), they potentially link the powerful geometric concept of Non-Uniform Rational B-Splines (NURBS) to meshes with T-junctions (referred as “hanging nodes” in the Finite Element context) and, hence, the well-established framework of adaptive mesh refinement. However, Buffa et al. (2010) have shown that T-meshes can induce linear dependent T-spline blending functions. This prohibits the use of T-splines as a basis for analytical purposes such as solving a partial differential equation. In particular, the mesh refinement algorithm presented by Sederberg et al. (2004) does not preserve analysis-suitability in general. This insight motivated the research on T-meshes that guarantee the linear independence of the corresponding T-spline blending functions, referred to as analysis-suitable T-meshes. Analysis-suitability has been characterized in terms of topological mesh properties in 2d (Li et al., 2012) and, in an alternative approach, through the equivalent concept of Dual-Compatibility (da Veiga et al., 2012), which allows for generalization to three-dimensional meshes.
