Splines over triangulations and splines over quadrangulations (tensor product splines) are two common ways to extend bivariate polynomials to splines. However, combination of both approaches leads to splines defined over mixed triangle and quadrilateral meshes using the isogeometric approach. Mixed meshes are especially useful for representing complicated geometries obtained e.g. from trimming. As (bi)-linearly parameterized mesh elements are not flexible enough to cover smooth domains, we focus in this work on the case of planar mixed meshes parameterized by (bi)-quadratic geometry mappings. In particular we study in detail the space of $C^1$-smooth isogeometric spline functions of general polynomial degree over two such mixed mesh elements. We present the theoretical framework to analyze the smoothness conditions over the common interface for all possible configurations of mesh elements. This comprises the investigation of the dimension as well as the construction of a basis of the corresponding $C^1$-smooth isogeometric spline space over the domain described by two elements. Several examples of interest are presented in detail.

翻译：基于三角剖分和四边形剖分（张量积样条）的样条是扩展二元多项式为样条的两种常见方法。然而，将这两种方法相结合，通过等几何方法可得到定义在三角形和四边形混合网格上的样条。混合网格特别适用于表示通过裁剪等操作获得的复杂几何形状。由于（双）线性参数化的网格单元不足以覆盖光滑区域，本文聚焦于采用（双）二次几何映射参数化的平面混合网格。我们详细研究了两个此类混合网格单元上一般多项式度的$C^1$光滑等几何样条函数空间。提出了分析所有可能网格单元配置下公共界面光滑性条件的理论框架，包括对维度的研究以及由两个单元描述的区域上相应$C^1$光滑等几何样条空间基函数的构造。文中详细给出了若干具有代表性的算例。