Construction of approximate C^1 bases for isogeometric analysis on two-patch domains
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In this paper, we develop and study approximately smooth basis constructions for isogeometric analysis over two-patch domains. One key element of isogeometric analysis is that it allows high order smoothness within one patch. However, for representing complex geometries, a multi-patch construction is needed. In this case, a C^0-smooth basis is easy to obtain, whereas C^1-smooth isogeometric functions require a special construction. Such spaces are of interest when solving numerically fourth-order PDE problems, such as the biharmonic equation and the Kirchhoff-Love plate or shell formulation, using an isogeometric Galerkin method. With the construction of so-called analysis-suitable G^1 (in short, AS-G^1) parametrizations, as introduced in (Collin, Sangalli, Takacs; CAGD, 2016), it is possible to construct C^1 isogeometric spaces which possess optimal approximation properties. These geometries need to satisfy certain constraints along the interfaces and additionally require that the regularity r and degree p of the underlying spline space satisfy 1 ≤ r ≤ p-2. The problem is that most complex geometries are not AS-G^1 geometries. Therefore, we define basis functions for isogeometric spaces by enforcing approximate C^1 conditions following the basis construction from (Kapl, Sangalli, Takacs; CAGD, 2017). For this reason, the defined function spaces are not exactly C^1 but only approximately. We study the convergence behavior and define function spaces that converge optimally under h-refinement, by locally introducing functions of higher polynomial degree and lower regularity. The convergence rate is optimal in several numerical tests performed on domains with non-trivial interfaces. While an extension to more general multi-patch domains is possible, we restrict ourselves to the two-patch case and focus on the construction over a single interface.
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