Bases and dimensions of C1-smooth isogeometric splines on volumetric two-patch domains

Abstract

We analyze the spaces of trivariate C1–smooth isogeometric functions on two-patch domains. Our aim is to generalize the corresponding results from the bivariate (Kapl et al. (2015) [25]) to the trivariate case. In the first part of the paper, we introduce the notion of gluing data and use it to define glued spline functions on two-patch domains. Applying the fundamental observation that “matched Gk–constructions always yield Ck–continuous isogeometric elements”, see Groisser and Peters (2015) [14], to graph hypersurfaces in four-dimensional space, allows us to characterize C1–smooth geometrically continuous isogeometric functions as the push-forwards of these functions for suitable gluing data. The second part of the paper is devoted to various special classes of gluing data. We analyze how the generic dimensions depend on the number of knot spans (elements) and on the spline degree. Finally we show how to construct locally supported basis functions in specific situations.