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：三角形网格上的样条与四边形网格上的样条（张量积样条）是两种将二元多项式扩展至样条的常用方法。然而，这两种方法的结合催生了基于等几何方法定义的、定义于三角形与四边形混合网格上的样条。混合网格在表示经裁剪等操作获得的复杂几何形状时尤为实用。由于（双）线性参数化网格单元不足以覆盖光滑区域，本文聚焦于由（双）二次几何映射参数化的平面混合网格情形。特别地，我们详细研究了在两个此类混合网格单元上、一般多项式次数的$C^1$光滑等几何样条函数空间。我们提出了分析所有可能网格单元配置下公共界面上光滑条件的理论框架，包括对应维度的探究以及由两个单元描述区域上$C^1$光滑等几何样条空间的基函数构造。文中详细给出了若干具有代表性的算例。