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.

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