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.

翻译：多元B样条与非均匀有理B样条（NURBS）因其张量积结构而缺乏自适应性。截断层次B样条（THB样条）为此提供了解决方案。THB样条将参数空间组织成层次结构，使得能够高效逼近和表示具有不同细节层次的函数。截断机制保证了B样条的单位分解性质，并在多层样条空间中定义了更分散、无重叠的一组基函数。将这些多层样条转换为Bézier单元表示，可简化其与现有有限元代码的整合。通过将THB样条的多层级提取与标准Bézier提取分离，可建立一种适用于任意嵌套空间序列的更通用独立框架。针对THB样条的多层级结构算子与Bézier提取算子均以局部方式构建。从单元视角调整多层级结构算子，并将其与这些单元的Bézier提取算子相乘，即可获得Bézier单元与层次结构之间的直接映射。具体实现中采用了名为GeoPDEs的开源Octave/MATLAB等几何分析代码。通过求解经典泊松问题，对比分析了多层级Bézier提取与标准THB样条方法的性能表现。