دانلود رایگان مقاله انالیز اصلاح تطبیقی T-مش با پیچیدگی خطی

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.

7. Conclusion

We presented an adaptive refinement algorithm for a subclass of analysis-suitable T-meshes that produces nested T-spline spaces, and we proved theoretical properties that are crucial for the analysis of adaptive schemes driven by a posteriori error estimators. As an example, compare Assumptions (2.9) and (2.10) from Carstensen et al. (2014) to Theorem 6.1 and Lemma 4.4, respectively. The presented refinement algorithm can be extended to the three-dimensional case, which is our current work. The factor Cp,q from the complexity estimate is affine in each of the parameters p, q and increases exponentially with growing dimension. We aim to apply the proposed algorithm to proof the rate-optimality of an adaptive algorithm for the numerical solution of second-order linear elliptic problems using T-splines as ansatz functions. Similar results have been proven for simple FE discretizations of the Poisson model problem by Stevenson (2007), Cascon et al. (2008), and recently for a wide range of discretizations and model problems by Carstensen et al. (2014).