Falai Chen

Ji Sheng, Xiaodong Wei, Falai Chen

This work presents a weighted quadrature (WQ) method to fast assemble Galerkin matrices based on unstructured spline surfaces. The method is developed upon a particular variant of unstructured splines, namely the bicubic analysis-suitable unstructured T-splines (ASUTS). While existing WQ approaches have significant speedup for structured splines (e.g., B-splines), their extension to unstructured splines faces several challenges: (1) lack of a global parametric domain for defining quadrature points, (2) a varying number of basis functions across elements that complicates the determination of the optimal number of quadrature points, and (3) ill-conditioned underdetermined linear systems that must be solved to find the quadrature weights. To solve these issues, we first define the WQ rule directly in the physical domain. Second, we specify the number of quadrature points function-wise (rather than element-wise), which naturally satisfies the well-posedness condition, namely the number of unknown weights no less than that of exactness constraints. Third, we employ the truncated Singular Value Decomposition to improve the conditioning of the underdetermined systems by discarding extremely small singular values, which are caused by the splines around extraordinary points. Several different model problems are studied, such as Poisson's problem, the biharmonic problem, and the nonlinear heat transfer problem. In the end, a variety of numerical tests are performed to demonstrate the accuracy and efficiency of the proposed method.

Jiansong Deng, Falai Chen, Liangbing Jin

This paper discusses the dimensions of the spline spaces over T-meshes with lower degree. Two new concepts are proposed: extension of T-meshes and spline spaces with homogeneous boundary conditions. In the dimension analysis, the key strategy is linear space embedding with the operator of mixed partial derivative. The dimension of the original space equals the difference between the dimension of the image space and the rank of the constraints which ensuring any element in the image space has a preimage in the original space. Then the dimension formula and basis function construction of bilinear spline spaces of smoothness order zero over T-meshes are discussed in detail, and a dimension lower bound of biquadratic spline spaces over general T-meshes is provided. Furthermore, using level structure of hierarchical T-meshes, a dimension formula of biquadratic spline space over hierarchical T-meshes are proved. A topological explantation of the dimension formula is shown as well.

Jing Li, Yihang Fu, Falai Chen

Boundary representation (B-rep) is the de facto standard for modern CAD, yet learning-based B-rep synthesis remains challenging due to the tight coupling between discrete topology and continuous geometry. We observe a fundamental asymmetry in B-reps: while wireframe composition involves high-entropy structural decisions, the interior surface geometry is largely constrained by its boundary loops. Motivated by this observation, we propose BrepForge, a generative framework that factorizes B-rep synthesis into two stages: wireframe composition and boundary-conditioned surface instantiation. In the first stage, a face-aware autoregressive model serializes the wireframe into structured sequences that explicitly encode hierarchical Vertex-Edge-Face (V-E-F) connectivity, yielding a topologically complete scaffold. In the second stage, precise surface geometries are instantiated by incorporating learning-free geometric priors derived from boundaries, transforming the complex synthesis task into a structured refinement process. This factorized approach ensures both topological integrity and geometric precision, effectively addressing the inherent complexities of B-rep modeling. Extensive experiments demonstrate that BrepForge outperforms existing baselines with superior geometric complexity and topological validity.

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.

Jing Li, Yihang Fu, Falai Chen

Boundary representation (B-rep) of geometric models is a fundamental format in Computer-Aided Design (CAD). However, automatically generating valid and high-quality B-rep models remains challenging due to the complex interdependence between the topology and geometry of the models. Existing methods tend to prioritize geometric representation while giving insufficient attention to topological constraints, making it difficult to maintain structural validity and geometric accuracy. In this paper, we propose DTGBrepGen, a novel topology-geometry decoupled framework for B-rep generation that explicitly addresses both aspects. Our approach first generates valid topological structures through a two-stage process that independently models edge-face and edge-vertex adjacency relationships. Subsequently, we employ Transformer-based diffusion models for sequential geometry generation, progressively generating vertex coordinates, followed by edge geometries and face geometries which are represented as B-splines. Extensive experiments on diverse CAD datasets show that DTGBrepGen significantly outperforms existing methods in both topological validity and geometric accuracy, achieving higher validity rates and producing more diverse and realistic B-reps. Our code is publicly available at https://github.com/jinli99/DTGBrepGen.

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.

Falai Chen, Xuhui Wang

In this paper we discuss the relationship between the moving planes of a rational parametric surface and the singular points on it. Firstly, the intersection multiplicity of several planar curves is introduced. Then we derive an equivalent definition for the order of a singular point on a rational parametric surface. Based on the new definition of singularity orders, we derive the relationship between the moving planes of a rational surface and the order of singular points. Especially, the relationship between the $μ$-basis and the order of a singular point is also discussed.