Coupling adaptively refined multi-patch spline discretizations via boundary compatibility

The present paper studies adaptive refinement on multi-patch structures via the gluing approach. More precisely, we investigate the applicability of the gluing construction in isogeometric analysis on multi-patch domains with adaptively refined spline spaces. We will see that this is closely related to the concept of boundary compatibility of an adaptive spline construction. Given a spline basis (or, more generally, a generating system if linear independence is not guaranteed) on a d-dimensional box domain, there are two possibilities for constructing the spline basis on the domain boundary. Firstly, one can simply restrict the basis functions to the boundary. Secondly, one may restrict the underlying mesh to the boundary and construct the spline basis on the resulting mesh. The two constructions do not necessarily produce the same set of functions. If they do, then the spline bases are said to be compatible. We study this property for hierarchical (HB-) and truncated hierarchical B-splines (THB-splines) and identify sufficient conditions. These conditions are weaker for THBthan for HB-splines. Finally we demonstrate the importance of boundary compatibility for geometric modeling and for adaptive refinement in isogeometric analysis, in particular when considering multi-patch domains.