Multi-degree splines are piecewise polynomial functions having sections of different degrees. They offer significant advantages over the classical uniform-degree framework, as they allow for modeling complex geometries with fewer degrees of freedom and, at the same time, for a more efficient engineering analysis. Moreover they possess a set of basis functions with similar properties to standard B-splines. In this paper we develop an algorithm for efficient evaluation of multi-degree B-splines, which, unlike previous approaches, is numerically stable. The proposed method consists in explicitly constructing a mapping between a known basis and the multi-degree B-spline basis of the space of interest, exploiting the fact that the two bases are related by a sequence of knot insertion and/or degree elevation steps and performing only numerically stable operations. In addition to theoretically justifying the stability of the algorithm, we will illustrate its performance through numerical experiments that will serve us to demonstrate its excellent behavior in comparison with existing methods, which, in some cases, suffer from apparent numerical problems.

多阶样条是具有不同阶段的分段多项式函数。与经典的均匀度框架相比，它们具有显着的优势，因为它们允许以较少的自由度对复杂的几何形状进行建模，同时可以进行更有效的工程分析。此外，它们拥有一组与标准 B 样条具有相似特性的基函数。在本文中，我们开发了一种有效评估多度 B 样条的算法，与以前的方法不同，该算法在数值上是稳定的。所提出的方法包括明确构建已知基和感兴趣空间的多度 B 样条基之间的映射，利用两个碱基通过一系列结插入和/或度数提升步骤相关的事实，并且仅执行数值稳定的操作。除了从理论上证明算法的稳定性之外，我们还将通过数值实验来说明其性能，这将有助于我们证明其与现有方法相比的出色行为，这些方法在某些情况下存在明显的数值问题。