We present an approximately $C^1$-smooth multi-patch spline construction which can be used in isogeometric analysis (IGA). A key property of IGA is that it is simple to achieve high order smoothness within a single patch. To represent more complex geometries one often uses a multi-patch construction. In this case, the global continuity for the basis functions is in general only $C^0$. Therefore, to obtain $C^1$-smooth isogeometric functions, a special construction for the basis is needed. Such spaces are of interest when solving numerically fourth-order problems, such as the biharmonic equation or Kirchhoff-Love plate/shell formulations, using an isogeometric Galerkin method. Isogeometric spaces that are globally $C^1$ over multi-patch domains can be constructed as in (Collin, Sangalli, Takacs; CAGD, 2016) and (Kapl, Sangalli, Takacs; CAGD, 2019). The constructions require so-called analysis-suitable $G^1$ parametrizations. To allow $C^1$ spaces over more general multi-patch parametrizations, we need to increase the polynomial degree and relax the $C^1$ conditions. We adopt the approximate $C^1$ construction for two-patch domains, as developed in (Weinm\"uller, Takacs; CMAME, 2021), and extend it to more general multi-patch domains. We employ the construction for a biharmonic model problem and compare the results with Nitsche's method. We compare both methods over complex multi-patch domains with non-trivial interfaces. The numerical tests indicate that the proposed construction converges optimally under $h$-refinement, comparable to the solution using Nitsche's method. In contrast to weakly imposing coupling conditions, the approximate $C^1$ construction is explicit and no additional terms need to be introduced to stabilize the method. Thus, the new proposed method can be used more easily as no parameters need to be estimated.
翻译:我们展示了大约$C1 $1 moot多端相位元的构造, 可用于等离子度分析( IGA) 。 IGA 的关键属性是, 在一个单一的补丁中, 实现高度秩序平稳很容易。 要代表更复杂的几何, 通常使用多端的构造。 在这种情况下, 基础功能的全球连续性一般只有$C1 美元。 因此, 要获得 $C1 moot 等离子函数, 需要为该基建立一个特殊的多级。 在解决数字四级问题时, 这种空间很有意义, 比如双相近的平价方程式或Kirchhoff 板块/ Shell配方, 使用一个等相近的Galerkin 方法。 在全球, $C%1 的多端域域的测量空间可以像( Collin, Sangalli, Takacs; CAGD, 2016) 和 (Kepl, Sang- suple, 和 CAGD, 2019) 那样需要如此的模型分析, $C- $CQQQQQQQQQQQQQQ1, 方法, 那样的平价方法, 那样, 。