Dimension Of Spline Space Research Articles

Counting the dimension of splines of mixed smoothness

In this paper, we study the dimension of bivariate polynomial splines of mixed smoothness on polygonal meshes. Here, “mixed smoothness” refers to the choice of different orders of smoothness across different edges of the mesh. To study the dimension of spaces of such splines, we use tools from homological algebra. These tools were first applied to the study of splines by Billera (Trans. Am. Math. Soc. 310(1), 325–340, 1988). Using them, estimation of the spline space dimension amounts to the study of the Billera-Schenck-Stillman complex for the spline space. In particular, when the homology in positions 1 and 0 of this complex is trivial, the dimension of the spline space can be computed combinatorially. We call such spline spaces “lower-acyclic.” In this paper, starting from a spline space which is lower-acyclic, we present sufficient conditions that ensure that the same will be true for the spline space obtained after relaxing the smoothness requirements across a subset of the mesh edges. This general recipe is applied in a specific setting: meshes of arbitrary topologies. We show how our results can be used to compute the dimensions of spline spaces on triangulations, polygonal meshes, and T-meshes with holes.

The Instability in the Dimensions of Spline Spaces Over T-Meshes with Nested T-Cycles

The Instability in the Dimensions of Spline Spaces Over T-Meshes with Nested T-Cycles

On the Problem of Instability in the Dimensions of Spline Spaces over T-Meshes with T-Cycles

The T-meshes are local modification of rectangular meshes which allow T-junctions. The splines over T-meshes are involved in many fields, such as finite element methods, CAGD etc. The dimension of a spline space is a basic problem for the theories and applications of splines. However, the problem of determining the dimension of a spline space is difficult since it heavily depends on the geometric properties of the partition. In many cases, the dimension is unstable. In this paper, we study the instability in the dimensions of spline spaces over T-meshes by using the smoothing cofactor-conformality method. The modified dimension formulas of spline spaces over T-meshes with T-cycles are also presented. Moreover, some examples are given to illustrate the instability in the dimensions of the spline spaces over some special meshes.

On the dimension of the bivariate spline space

The problem of determining the dimension of the bivariate spline space is considered. It is shown that for a large class of triangulations the dimension of the spline space is equal to Schumaker's lower bound. An algorithm to determine whether a given triangulation ▵ belongs to such a class is presented.

Local Dimension of Piecewise Polynomial Spaces, Syzygies, and Solutions of Systems of Partial Differential Equations

In this paper we discuss the relationship between syzygies, the dimension of spline spaces and the solutions of systems of differential equations. Several applications of our results are given.