Dimension and basis construction for $C^{2}$-smooth isogeometric spline spaces over bilinear-like $G^{2}$ two-patch parameterizations
For researchers in isogeometric analysis, this provides a constructive basis for C^2-smooth splines over a broader class of two-patch geometries, enabling higher-order smoothness in applications.
The paper introduces bilinear-like G^2 two-patch geometries, a wider class than bilinear parameterizations, and constructs a basis for C^2-smooth isogeometric spline spaces over them. The basis functions have simple closed forms, local supports, and good conditioning, with a subspace achieving optimal L^2-approximation properties.
A particular class of planar two-patch geometries, called bilinear-like $G^{2}$ two-patch geometries, is introduced. This class includes the subclass of all bilinear two-patch parameterizations and possesses similar connectivity functions along the patch interface. It is demonstrated that the class of bilinear-like $G^2$ two-patch parameterizations is much wider than the class of bilinear parameterizations and can approximate with good quality given generic two-patch parameterizations. We investigate the space of $C^{2}$-smooth isogeometric functions over this specific class of two-patch geometries. The study is based on the equivalence of the $C^2$-smoothness of an isogeometric function and the $G^2$-smoothness of its graph surface (cf. [12, 20]). The dimension of the space is computed and an explicit basis construction is presented. The resulting basis functions possess simple closed form representations, have small local supports, and are well-conditioned. In addition, we introduce a subspace whose basis functions can be generated uniformly for all possible configurations of bilinear-like $G^{2}$ two-patch parameterizations. Numerical results obtained by performing $L^{2}$-approximation indicate that already the subspace possesses optimal approximation properties.