Paper Info
Title
Polynomial Spline Spaces Of Non-Uniform Bi-Degree On T-Meshes: Combinatorial Bounds On The Dimension
Abstract
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.
Year
DOI
Venue
2021
10.1007/s10444-020-09829-4
ADVANCES IN COMPUTATIONAL MATHEMATICS
Keywords
DocType
Volume
Smooth splines, T-Meshes, Non-uniform degrees, Dimension formula, Homological algebra, 14Q99, 13P25, 68W30, 65D07, 65D17
Journal
47
Issue
ISSN
Citations 
1
1019-7168
0
PageRank 
References 
Authors
0.34
6
3
Name
Order
Citations
PageRank
Deepesh Toshniwal100.34
Bernard Mourrain21074113.70
Thomas J. R. Hughes320825.49