  Multivariate B-splines and Non-uniform rational B-splines (NURBS) lack
adaptivity due to their tensor product structure. Truncated hierarchical
B-splines (THB-splines) provide a solution for this. THB-splines organize the
parameter space into a hierarchical structure, which enables efficient
approximation and representation of functions with different levels of detail.
The truncation mechanism ensures the partition of unity property of B-splines
and defines a more scattered set of basis functions without overlapping on the
multi-level spline space. Transferring these multi-level splines into B\'ezier
elements representation facilitates straightforward incorporation into existing
finite element (FE) codes. By separating the multi-level extraction of the
THB-splines from the standard B\'ezier extraction, a more general independent
framework applicable to any sequence of nested spaces is created. The operators
for the multi-level structure of THB-splines and the operators of B\'ezier
extraction are constructed in a local approach. Adjusting the operators for the
multi-level structure from an element point of view and multiplying with the
B\'ezier extraction operators of those elements, a direct map between B\'ezier
elements and a hierarchical structure is obtained. The presented implementation
involves the use of an open-source Octave/MATLAB isogeometric analysis (IGA)
code called GeoPDEs. A basic Poisson problem is presented to investigate the
performance of multi-level B\'ezier extraction compared to a standard
THB-spline approach.
