Multi-level Bézier extraction of truncated hierarchical B-splines for isogeometric analysis

TL;DR

This work addresses adaptivity in isogeometric analysis by employing truncated hierarchical B-splines (THB-splines) for local refinement and introducing a generalized multi-level Bézier extraction framework that maps hierarchical spline spaces to Bézier elements. By constructing global and local extraction operators and composing them with the standard Bézier extractor, the method provides a direct Bernstein-based representation per Bézier element compatible with conventional FE solvers. The approach preserves key spline properties such as locality and partition of unity and matches the accuracy of local THB refinements, demonstrated on a Poisson problem in a unit square. Overall, the paper offers a practical, general pathway to integrate adaptive THB-splines into existing FE workflows and extends naturally to other nested spaces.

Abstract

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 Bézier elements representation facilitates straightforward incorporation into existing finite element (FE) codes. By separating the multi-level extraction of the THB-splines from the standard Bézier 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 Bézier 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 Bézier extraction operators of those elements, a direct map between Bézier 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 Bézier extraction compared to a standard THB-spline approach.

Figures (7)

• Figure 1: Basis functions of a univariate quadratic B-spline and the corresponding curve. $C^0$ continuity at the $knot = 0.75$.
• Figure 2: Basis functions for the computational domain. The gray functions are the deactivated ones and the colored ones are the active ones. The $N_3^{(0)}$ and $N_7^{(1)}$ are truncated functions.
• Figure 3: Translate from hierarchical spline space to local Bernstein polynomials.
• Figure 4: Newton-Cotes integration points appear in the boundary of the elements and can not be evaluated correctly due to the Cox-de Boor formula. Bézier extraction, in the right figure, overcomes this limitation with a standard set of Bernstein polynomials.
• Figure 5: Exact solution of the unit square boundary value problem.