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 $1 \leq r \leq p-2$. 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.

中文翻译：

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构造用于两个面片域的等几何分析的近似$ C ^ 1 $基数

在本文中，我们开发和研究了两个面片区域上等几何分析的近似光滑基础构造。等几何分析的一个关键要素是，它允许在一个面片内实现高阶平滑度。然而，为了表示复杂的几何形状，需要多面体构造。在这种情况下，容易获得$ C ^ 0 $-平滑的基数，而$ C ^ 1 $-平滑的等几何函数需要特殊的构造。当使用等几何Galerkin方法求解数值四阶PDE问题（例如双调和方程和Kirchhoff-Love板或壳公式）时，此类空间是令人关注的。如（Collin，Sangalli，Takacs; CAGD，2016）所述，通过构建所谓的适合分析的$ G ^ 1 $（简称AS- $ G ^ 1 $）参数化，可以构造具有最佳逼近特性的$ C ^ 1 $等几何空间。这些几何形状需要满足沿着界面的某些约束，并且还需要基础样条空间的正则性$ r $和度数$ p $满足$ 1 \ leq r \ leq p-2 $。问题在于，大多数复杂的几何图形都不是AS- $ G ^ 1 $几何图形。因此，我们根据（Kapl，Sangalli，Takacs; CAGD，2017）的基础构造，通过执行近似$ C ^ 1 $条件来定义等几何空间的基础函数。由于这个原因，定义的函数空间不完全是$ C ^ 1 $，而仅仅是大约$ C ^ 1 $。我们通过局部引入较高多项式度和较低正则性的函数，研究收敛行为并定义在$ h $ -refine下最优收敛的函数空间。在具有非平凡接口的域上执行的几个数值测试中，收敛速度是最佳的。虽然可以扩展到更通用的多修补程序域，但我们将自己限制在两个修补程序的情况下，并专注于通过单个接口进行构造。