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.
Highlights
Standard B-spline parameterizations of surfaces in computer-aided geometric design are defined on a grid of nodes over a rectangular domain
In this paper, motivated by applications for isogeometric finite element methods, we study the space of piecewise polynomials functions on a T-mesh with different bi-degrees on its faces and different regularities across its edges
Given a constructive approach that aims to produce linearly independent splines over T-meshes using only local data, computation of the associated spline space dimension can help identify cases where the splines produced by the approach cannot be linearly independent
Summary
Standard B-spline parameterizations of surfaces in computer-aided geometric design are defined on a grid of nodes over a rectangular domain. The construction of so-called LR-splines defined on T-meshes and based on knot sub-grids has been proposed in [12] Their use in isogeometric analysis has been further investigated in [17], including an analysis of the linear independency of the blending functions [6]. Since any efficient constructive approach must rely only on local data for building spline functions, this instability in the dimension necessitates identification of configurations where the spline space dimension is a priori guaranteed to be stable In this direction, a detailed study of spline spaces on general T-meshes has been presented in [22] using homological techniques, which go back to [2]. Given a constructive approach that aims to produce linearly independent splines over T-meshes using only local data, computation of the associated spline space dimension can help identify cases where the splines produced by the approach cannot be linearly independent
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