Dimension of Bi-degree $(d,d)$ Spline Spaces with the Highest Order of Smoothness over Hierarchical T-Meshes

TL;DR

The paper addresses the problem of determining the dimension of the bi-degree $(d,d)$ spline space $S_d(\mathscr{T})$ with the highest order of smoothness over hierarchical T-meshes. It introduces the tensor product T-connected component and develops a smoothing cofactor-conformality framework to compute the dimension of the conformality vector space (CVS) recursively level by level under tensor-product subdivisions. A closed-form dimension formula for $S_d(\mathscr{T})$ is derived in terms of vertex/edge counts, and the authors establish a CVR-graph equivalence that connects the spline space to a lower-degree space on the CVR graph, enabling basis construction strategies. They further provide a mesh-modification approach to ensure dimension stability and demonstrate the CVS-CVR correspondence under cross subdivision, extending prior results to general $d$ and paving the way for practical basis design in isogeometric analysis. The results unify hierarchical T-mesh spline theory with CVR-graph techniques, offering robust dimension computation and constructive pathways for basis functions.

Abstract

In this article, we study the dimension of the spline space of di-degree $(d,d)$ with the highest order of smoothness over a hierarchical T-mesh $\mathscr T$ using the smoothing cofactor-conformality method. Firstly, we obtain a dimensional formula for the conformality vector space over a tensor product T-connected component. Then, we prove that the dimension of the conformality vector space over a T-connected component of a hierarchical T-mesh under the tensor product subdivision can be calculated in a recursive manner. Combining these two aspects, we obtain a dimensional formula for the bi-degree $(d,d)$ spline space with the highest order of smoothness over a hierarchical T-mesh $\mathscr T$ with mild assumption. Additionally, we provide a strategy to modify an arbitrary hierarchical T-mesh such that the dimension of the bi-degree $(d,d)$ spline space is stable over the modified hierarchical T-mesh. Finally, we prove that the dimension of the spline space over such a hierarchical T-mesh is the same as that of a lower-degree spline space over its CVR graph. Thus, the proposed solution can pave the way for the subsequent construction of basis functions for spline space over such a hierarchical T-mesh.

Figures (12)

• Figure 1: A T-mesh and its T-connected component
• Figure 2: A hierarchical T-mesh and T-connected component in each level.
• Figure 3: A T-mesh and its extended T-mesh for $S_3(\mathscr{T})$.
• Figure 4: Local conformality condition
• Figure 5: global conformality condition along a horizontal T $l$-edge

Theorems & Definitions (37)

• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Theorem 2.1
• Definition 2.5
• Proposition 2.1
• Definition 2.6
• Definition 2.7
• Proposition 2.2