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Polynomial spline spaces of non-uniform bi-degree on T-meshes: combinatorial bounds on the dimension
Advances in Computational Mathematics ( IF 1.7 ) Pub Date : 2021-02-02 , DOI: 10.1007/s10444-020-09829-4
Deepesh Toshniwal , Bernard Mourrain , Thomas J. R. Hughes

Polynomial splines are ubiquitous in the fields of computer-aided geometric design and computational analysis. Splines on T-meshes, especially, have the potential to be incredibly versatile since local mesh adaptivity enables efficient modeling and approximation of local features. Meaningful use of such splines for modeling and approximation requires the construction of a suitable spanning set of linearly independent splines, and a theoretical understanding of the spline space dimension can be a useful tool when assessing possible approaches for building such splines. Here, we provide such a tool. Focusing on T-meshes, we study the dimension of the space of bivariate polynomial splines, and we discuss the general setting where local mesh adaptivity is combined with local polynomial degree adaptivity. The latter allows for the flexibility of choosing non-uniform bi-degrees for the splines, i.e., different bi-degrees on different faces of the T-mesh. In particular, approaching the problem using tools from homological algebra, we generalize the framework and the discourse presented by Mourrain (Math. Comput. 83(286):847–871, 2014) for uniform bi-degree splines. We derive combinatorial lower and upper bounds on the spline space dimension and subsequently outline sufficient conditions for the bounds to coincide.



中文翻译:

T网格上非均匀双度的多项式样条空间:维上的组合界

多项式样条在计算机辅助几何设计和计算分析领域无处不在。特别是T形网格上的样条曲线具有无限的通用性,因为局部网格自适应性可实现高效的局部特征建模和近似。要在建模和逼近中有意义地使用此类样条曲线,就需要构建合适的线性独立样条曲线跨度集合,并且在评估构建此类样条曲线的可能方法时,对样条曲线空间尺寸的理论理解可能是有用的工具。在这里,我们提供了这样的工具。着眼于T网格,我们研究了二元多项式样条的空间维,并讨论了局部网格自适应性与局部多项式适应性相结合的一般设置。后者允许为样条曲线选择非均匀的双向度,即在T网格的不同面上具有不同的双向度。尤其是,使用同源代数的工具来解决问题时,我们对Mourrain(Math。Comput。83(286):847–871,2014年),用于均匀的双度样条。我们导出样条空间维数上的组合下界和上限,并随后概述边界一致的充分条件。

更新日期:2021-02-02
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