arXiv subjects

Bingru Huang

Publications and source records attributed to Bingru Huang.

8 recordsLinked to original sources

arXiv2026

PH2T-splines, Part I: A Reasonable Mesh Assumption

This paper is the first in a three-part series on the construction of polynomial splines with the highest order of smoothness over hierarchical T-meshes, referred to as $\PHtwoT$-splines. For splines of bi-degree $(d,d)$, we study suitable refinement conditions for the subsequent basis construction, which requires dimensional stability of the underlying spline space. We present two groups of examples, considering unrestricted hierarchical refinement and refinement without vanishable T $l$-edges, respectively. The first setting permits new edges without additional degrees of freedom. In the second group, every refinement level excludes vanishable T $l$-edges and increases the dimension. Nevertheless, the dimension is unstable in both groups. We then introduce template translations to describe each refined region as a union of translates of a fixed template in the cell-index grid. Together with the known stability result under $(d-1)\times(d-1)$ template refinement, these examples justify this condition as a reasonable mesh assumption for the subsequent $\PHtwoT$-spline construction.

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arXiv2026

Sharp Dimension Bounds for Spline Spaces over T-meshes with Highest Order of Smoothness

The dimension of a polynomial spline space of bi-degree $(d_1,d_2)$ over a T-mesh $\mathscr{T}$ with the highest order of smoothness $(d_1-1,d_2-1)$ depends on both mesh topology and geometric configurations. Under the assumption that the T-connected components of the T-mesh $\mathscr{T}$ contain no vanishable T $l$-edges, we develop explicit upper and lower bounds of the dimension of the polynomial spline space. By introducing a decoupling technique within the completely non-diagonalizable component (CNDC) of the T-mesh $\mathscr{T}$, we separate tightly coupled multi-vertex constraints and transform global conformality conditions into localized linear equations along each interior large edge. Based on the decoupling technique, a new dimension formula of the polynomial spline space is then presented, and from which sharp upper and lower bounds of the dimension are obtained. The bounds are sharp in the sense that different geometric realizations of T-meshes with the same topology can attain the lower and upper bounds for the dimension of the polynomial spline space. We further prove that the new formula is consistent with Mourrain's homological dimension formula, and a sharper lower bound is obtained by our method.

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arXiv2026

A Homological Decomposition for the Dimension and Dimensional Stability of Polynomial Spline Spaces over T-Meshes

We study the dimension and dimensional stability of polynomial spline spaces of bi-degree $(m,m')$ with prescribed smoothness orders over planar T-meshes. The homological dimension formula writes the spline dimension as the sum of an Euler characteristic term and a correction term. For a fixed ordered bi-degree, the Euler characteristic term is determined by the mesh structure and the prescribed smoothness orders. The correction term can be written as a quotient of coefficient spaces attached to maximal interior segments (MISs). We prove a weighted deletion theorem: when the available vertex relations generate the coefficient space of an MIS, its summand can be removed from the quotient without changing the correction term. This operation changes neither the T-mesh nor its chain complexes. Repeating the deletion leaves a weighted completely non-diagonalizable component (CNDC). We prove that the weighted CNDC is independent of the order in which eligible MISs are removed and is the same for corresponding pairs in the structural class. The correction term can therefore be represented using only the MISs in the weighted CNDC, while all relations from the original T-mesh are retained. Dimensional stability is then equivalent to constancy of the dimension of the remaining relation space. In particular, an empty weighted CNDC is sufficient for stability. We also derive an upper bound for the remaining correction term and compare it with Mourrain's upper bound based on all MISs.

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arXiv2026

Dimension Calculation for Spline Spaces over Rectilinear Partitions via Smoothing Cofactor Method

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.

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arXiv2025

Basis Construction for Spline Spaces over Arbitrary Partitions from a Dimensional Stable Perspective

This paper introduces a novel framework for constructing $C^r$ basis functions for polynomial spline spaces of degree $d$ over arbitrary planar polygonal partitions, overturning the belief that basis functions cannot be constructed on dimensionally unstable meshes. We provide a comprehensive comparison of basis construction methods, classifying them as explicit, semi-explicit, and implicit. Our method, a semi-explicit construction using Extended Edge Elimination conditions, uniquely resolves all theoretical challenges in spline spaces by ensuring a complete basis. For the first time, we construct basis functions for the spline space over the Morgan-Scott partition, previously unachieved, and elucidate dimensional instability through this construction.

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arXiv2025

Basis construction for polynomial spline spaces over arbitrary T-meshes

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.

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arXiv2025

A Preliminary Study on the Dimensional Stability Classification of Polynomial Spline Spaces over T-meshes

This paper studies dimensional stability of polynomial spline spaces over T-meshes with highest-order smoothness. Dimensional stability is defined as the invariance of the dimension of the spline space over structurally isomorphic T-meshes, where a structurally isomorphic map preserves edge intersections and the mutual positions of all edges. Absolute stability, a stronger notion, is introduced via structurally similar maps that depend solely on the topology of T-connected components. It is proved that every T-connected component admits a unique decomposition into a diagonalizable part and a completely non-diagonalizable component (CNDC). This reduces stability to the rank stability of the conformality matrix associated with the CNDC. For diagonalizable T-meshes, the conformality vector space decomposes as a direct sum over independent T $l$-edges, supporting basis function construction per T $l$-edge. These results provide a systematic framework for classifying dimensional stability and a rigorous theoretical foundation for constructing basis functions of spline spaces over T-meshes.

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arXiv2025

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

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.

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