Modified Bases of PHT-Splines
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Modified Bases of PHT-Splines Yuanpeng Zhu1 · Falai Chen1
Received: 24 January 2017 / Revised: 19 April 2017 / Accepted: 12 July 2017 / Published online: 7 September 2017 © School of Mathematical Sciences, University of Science and Technology of China and Springer-Verlag GmbH Germany 2017
Abstract Recently, it was found that during the process of certain refinement of hierarchical T-meshes, some basis functions of PHT-splines decay severely, which is not expected in solving numerical PDEs and in least square data fitting since the matrices assembled by these basis functions are likely to be ill-conditioned. In this paper, we present a method to modify the basis functions of PHT-splines in the case that the supports of the original truncated basis functions are rectangular domains to overcome the decay problem. The modified basis functions preserve the same nice properties of the original PHT-spline basis functions such as partition of unity, local support, linear independency. Numerical examples show that the modified basis functions can greatly decrease the condition numbers of the stiffness matrices assembled in solving Poisson’s equation with Dirichlet boundary conditions. Keywords PHT-spline · B-spline · Hierarchical T-mesh · Poisson’s equation Mathematics Subject Classification 65D07 · 65D17
1 Introduction Locally refinable splines have been a hot topic in computer-aided geometric design (CAGD) and isogeometric analysis (IGA) in recent years. The motivation to apply such
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Falai Chen [email protected] Yuanpeng Zhu [email protected]
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School of Mathematical Sciences, University of Science and Technology of China, Hefei 230026, People’s Republic of China
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Y. Zhu, F. Chen
splines in design and analysis is to overcome the bottleneck of tensor product structure of NURBS. By allowing T-junctions in the control meshes, T-splines were proposed by Sederberg and his colleagues which have shown great potential applications in geometric modeling and adaptive IGA [1–3]. However, the blending functions of T-splines may be linearly dependent as pointed out by Buffa et al. [4]. In order to fix the linear dependency of T-spline blending functions, analysis suitable T-splines (AST-splines for short) were proposed [5,6]. AST-splines form a subset of T-splines and possess desired properties, such as partition of unity, local support and linearly independent, which are vital for design and analysis [7]. However, local refinement in AST-splines may propagate beyond the domain of interest. To address the problem of local refinement of splines on rectangular domain, the concept of hierarchical B-splines (HB-splines for short) was firstly introduced by Forsey and Bartels as an accumulation of B-splines with nested knot vectors [8]. HB-splines can be locally refined using overlays. In [9], Giannelli et al. normalized HB-splines by using a truncation mechanism to reduce the support of basis functions. They call such HB-splines as truncated hierarchical B-splines (THB-splines for short). The basis functions of THB-splin
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