Efficient extraction of hierarchical B-Splines for local refinement and coarsening of Isogeometric Analysis

The main motivation for hierarchical B-Splines in Isogeometric Analysis is to perform a local refinement that is globally not a tensor product. Compared to standard knot insertion, this allows to increase refinement locality. Moreover, the implementation of hierarchical refinement on existing codes can be facilitated by Bézier extraction. This approach was initially developed for unrefined patches and then extended to local refinement. In case of B-Spline patches that are not locally refined, extraction and projection algorithms can be optimized by leveraging the tensor structure. However, these optimizations are not straightforwardly applicable to hierarchical B-Splines. In this contribution, we show an approach where all extraction operations are formulated to work directly on the univariate extraction operators, without explicitly computing and storing the full tensor product. In particular, we consider the extraction of functions, and projection of degrees of freedom for refinement and coarsening. A slightly modified Bézier projection is proposed to fully exploit the developed algorithms. We finally compare our approach to an explicit computation of the extraction operator in both a linear transient and a nonlinear example.