Extraction of Hierarchical B-Splines

Non-intrusive and efficient implementation of hierarchical B-spline refinement for Isogeometric Analysis


One of the main topics of research on Isogeometric Analysis is local refinement. Among the various techniques currently studied and developed, one of the most appealing, referred to as hierarchical B-Splines, consists of defining a suitable set of basis functions on different hierarchical levels. This strategy can also be improved, for example to recover partition of unity, resorting to a truncation operation,  giving rise to the so-called truncated hierarchical B-Splines. Despite its conceptual simplicity, implementing the hierarchical definition of shape functions into an existing code can be rather involved.




In this project we present a simple way to bring the hierarchical isogeometric concept closer to a standard finite element formulation. Practically speaking, the hierarchy of functions and knot spans is flattened into a sequence of elements being equipped with a standard single-level basis. In fact, the proposed multi-level extraction is a generalization of the classical Bézier extraction and analogously offers a standard element structure to the hierarchical overlay of functions. Moreover, this approach is suitable for an extension to non-linear problems and for a parallel implementation.


In particular, basis functions local to one element can be represented as linear combination of standard B-splines of the level of the element under consideration. Such linear operation can be represented by a matrix multiplication. Moreover, by using the original (single-level) Bézier extraction, these standard B-splines can be locally expressed as linear combination of Bernstein polynomials. As a result, we obtain a direct map from a standard set of reference basis functions equal for each element, to the hierarchical local basis. Therefore, hierarchical refinements can be treated in a way that is very similar to standard finite element implementations.




In conclusion, this technique eases the integration of hierarchical refinement in existing codes, allowing code re-use and faster development.



fachinotti reduced


pynched reduced



[1] Davide D’Angella, Stefan Kollmannsberger, Ernst Rank, Alessandro Reali, Multi-level Bézier extraction for hierarchical local refinement of Isogeometric Analysis, Computer Methods in Applied Mechanics and Engineering, Volume 328, 2018, pp. 147-174, https://doi.org/10.1016/j.cma.2017.08.017.

Contact person

Davide D'Angella