Bingru Huang

Bingru Huang, Falai Chen

This paper presents a general framework for calculating the dimension of spline spaces over arbitrary rectilinear partitions using the smoothing cofactor method. The approach extends existing dimension theory for polynomial splines over T-meshes by introducing the concept of TE-connected components, reducing the problem to the rank computation of explicitly constructible conformality matrices. Furthermore, a new class of rectilinear partitions, termed partitions with disjoint truncated l-edges, is introduced. It is proven that under specific conditions, the dimension of the corresponding spline space attains Schumaker's lower bound. This shows that the lower bound is attainable for arbitrary degree d and smoothness order mu in certain partition configurations. Numerical examples, including the Morgan-Scott and Yuan-Stillman partitions, validate the effectiveness and generality of the framework for both triangular and non-triangular rectilinear partitions.

Shicong Zhong, Falai Chen, Bingru Huang

This paper presents the first method for constructing bases for polynomial spline spaces over an arbitrary T-meshes (PT-splines for short). We construct spline basis functions for an arbitrary T-mesh by first converting the T-mesh into a diagonalizable one via edge extension, ensuring a stable dimension of the spline space. Basis functions over the diagoalizable T-mesh are constructed according to the three components in the dimension formula corresponding to cross-cuts, rays, and T $l$-edges in the diagonalizable T-mesh, and each component is assigned some local tensor product B-splines as the basis functions. We prove this set of functions constitutes a basis for the diagonalizable T-mesh. To remove redundant edges from extension, we introduce a technique, termed Extended Edge Elimination (EEE) to construct a basis for an arbitrary T-mesh while reducing structural constraints and unnecessary refinements. The resulting PT-spline basis ensures linear independence and completeness, supported by a dedicated construction algorithm. A comparison with LR B-splines, which may lack linear independence and are limited to LR-meshes, highlights the PT-spline's versatility across any T-mesh. Examples are also provided to demonstrate that dimensional instability in spline spaces is related with basis function degradation and that PT-splines are advantageous over HB-splines for certain hierarchical T-meshes.

Bingru Huang, Falai Chen

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.