Multi-level B\'{e}zier extraction of truncated hierarchical B-splines for isogeometric analysis

TL;DR

This paper introduces a multi-level Bézier extraction method for truncated hierarchical B-splines, enhancing adaptivity and integration into finite element codes for isogeometric analysis.

Contribution

It develops a general, independent framework for multi-level B-spline extraction applicable to nested spaces, improving the efficiency of isogeometric analysis.

Findings

The method enables straightforward incorporation into existing FE codes.

Performance improvements over standard THB-splines demonstrated in a Poisson problem.

Implementation using open-source GeoPDEs code confirms effectiveness.

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 B\'{e}zier 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\'{e}zier 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\'{e}zier 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\'{e}zier extraction operators of those elements, a direct map between B\'{e}zier 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\'{e}zier extraction compared to a standard THB-spline approach.