arXiv subjects

Falai Chen

Publications and source records attributed to Falai Chen.

At least 19 recordsLinked to original sources

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.

math.NA↗

arXiv2026

Implicitization of rational hypersurfaces by syzygies with respect to coefficient ideals

We study rational hypersurfaces $\mathscr{S}$ defined as the closure of the image of a generically finite rational map $\phi:\mathscr{X}\rightarrow \mathbb{P}^{n+1}$, where $\mathscr{X}$ is an $n$-dimensional toric variety. We provide matrix representations for the implicitization of $\mathscr{S}$ that are constructed from the coefficients of linear syzygies and quadratic syzygies of the parametric equations. A central feature of our construction is the restriction of all coefficients in the Cox ring $R$ to a specific coefficient ideal $J$. In the two-dimensional case, this approach eliminates the need for $\phi$ to be locally a complete intersection at the base points, that is, the determinant of the implicitization matrix is equal to a power of the implicit equation for arbitrary base points. This result generalizes several previous results in surface implicitization.

math.AC↗

arXiv2026

Weighted Quadrature on Unstructured Splines

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.

math.NA↗

arXiv2026

BrepForge: Factorized B-rep Synthesis via Wireframe Composition and Boundary-Conditioned Surface Instantiation

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.

cs.GR↗

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.

math.NA↗

arXiv2026

An inertial minimal-deformation-rate framework for shape optimization

We propose a robust numerical framework for PDE-constrained shape optimization and Willmore-driven surface hole filling. To address two central challenges -- slow progress in flat energy landscapes, which can trigger premature stagnation at suboptimal configurations, and mesh deterioration during geometric evolution -- we couple a second-order inertial flow with a minimal-deformation-rate (MDR) mesh motion strategy. This coupling accelerates convergence while preserving mesh quality and thus avoids remeshing. To further enhance robustness for non-smooth or non-convex initial geometries, we incorporate surface-diffusion regularization within the Barrett-Garcke-N"urnberg (BGN) framework. Moreover, we extend the inertial MDR methodology to Willmore-type surface hole filling, enabling high-order smooth reconstructions even from incompatible initial data. Numerical experiments demonstrate markedly faster convergence to lower original objective values, together with consistently superior mesh preservation throughout the evolution.

math.NA↗

arXiv2026

Element-based B-spline basis function spaces: construction and application in isogeometric analysis

This paper develops a unified theoretical framework for constructing B-spline basis function spaces with structural equivalence to finite element spaces. The theory rigorously establishes that these bases emerge as explicit linear combinations of B-spline element bases. For any prescribed smoothness requirements, this element-wise formulation enables the Hermite interpolation at nodes, which directly utilizes function values and derivatives without solving global linear systems. By focusing on explicit interpolation properties, element-wise analysis establishes optimal approximation errors, even when the space smoothness attains its theoretical maximum for the space degree. In isogeometric analysis (IgA), the construction naturally decomposes geometric mappings into element-level representations, allowing efficient computations across elements regardless of node distribution. Notably, the same Hermite interpolation framework simultaneously handles domain parameterization and IgA solutions, allowing direct imposition of boundary conditions through function and derivative matching. Numerical tests demonstrate optimal convergence rates and superconvergence properties in 2D IgA under uniform knot configurations, and improved computational efficiency in 3D IgA with non-uniform knot distributions.

math.NA↗

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.

math.NA↗

arXiv2025

Polynomial preserving recovery for PHT-splines

We propose a polynomial preserving recovery method for PHT-splines within isogeometric analysis to obtain more accurate gradient approximations. The method fully exploits the local interpolation properties of PHT-splines and avoids the need for information on gradient superconvergent points. By leveraging the superconvergence argument of difference quotients and the interior error estimate, we establish the superconvergence property of the recovered gradient on translation invariant meshes. As a byproduct, a recovery-based a posteriori error estimator is developed for adaptive refinement. Numerical results confirm the theoretical findings and demonstrate the effectiveness of the proposed method.

math.NA↗

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.

math.NA↗

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.

math.NA↗

arXiv2025

DTGBrepGen: A Novel B-rep Generative Model through Decoupling Topology and Geometry

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.

cs.CV↗

arXiv2023

Adaptive Isogeometric Topology Optimization of Shell Structures based on PHT-splines

This paper proposes an Adaptive Isogeometric Topology Optimization framework for shell structures based on PHT-splines (PHT-AITO). In this framework, the design domain, displacement, and density are represented by PHT-splines. Leveraging the local refinement capability of PHT-splines, mesh elements defining the density function are adaptively refined to achieve a suitable resolution at the interface between solid and void regions. This addresses the issue of excessive degrees of freedom resulting from global refinement. The refinement of the mesh elements is driven by their density. During the optimization of the density on a refined mesh, the initial value of the density is inherited from the optimization results on the previous mesh to accelerate the iteration process and maintain the stability of the optimized structure. Numerical experiments on various shell structures have verified the effectiveness of PHT-AITO. Compared with isogeometric topology optimization based on tensor-product splines, PHT-AITO can significantly reduce the degrees of freedom in the optimization problem, thereby improving computational efficiency.

math.OC↗

arXiv2023

Density-based isogeometric topology optimization of shell structures

Shell structures with a high stiffness-to-weight ratio are desirable in various engineering applications. In such scenarios, topology optimization serves as a popular and effective tool for shell structures design. Among the topology optimization methods, solid isotropic material with penalization method(SIMP) is often chosen due to its simplicity and convenience. However, SIMP method is typically integrated with conventional finite element analysis(FEA) which has limitations in computational accuracy. Achieving high accuracy with FEA needs a substantial number of elements, leading to computational burdens. In addition, the discrete representation of the material distribution may result in rough boundaries and checkerboard structures. To overcome these challenges, this paper proposes an isogeometric analysis(IGA) based SIMP method for optimizing the topology of shell structures based on Reissner-Mindlin theory. We use NURBS to represent both the shell structure and the material distribution function with the same basis functions, allowing for higher accuracy and smoother boundaries. The optimization model takes compliance as the objective function with a volume fraction constraint and the coefficients of the density function as design variables. The Method of Moving Asymptotes is employed to solve the optimization problem, resulting in an optimized shell structure defined by the material distribution function. To obtain fairing boundaries in the optimized shell structure, further process is conducted by fitting the boundaries with fair B-spline curves automatically. Furthermore, the IGA-SIMP framework is applied to generate porous shell structures by imposing different local volume fraction constraints. Numerical examples are provided to demonstrate the feasibility and efficiency of the IGA-SIMP method, showing that it outperforms the FEA-SIMP method and produces smoother boundaries.

math.OC↗

arXiv2020

Determinantal tensor product surfaces and the method of moving quadrics

A tensor product surface $\mathscr{S}$ is an algebraic surface that is defined as the closure of the image of a rational map $ϕ$ from $\mathbb{P}^1\times \mathbb{P}^1$ to $\mathbb{P}^3$. We provide new determinantal representations of $\mathscr{S}$ under the assumptions that $ϕ$ is generically injective and its base points are finitely many and locally complete intersections. These determinantal representations are matrices that are built from the coefficients of linear relations (syzygies) and quadratic relations of the bihomogeneous polynomials defining $ϕ$. Our approach relies on a formalization and generalization of the method of moving quadrics introduced and studied by David Cox and his co-authors.

math.AG↗

arXiv2019

Volumetric Spline Parameterization for Isogeometric Analysis

Given the spline representation of the boundary of a three dimensional domain, constructing a volumetric spline parameterization of the domain (i.e., a map from a unit cube to the domain) with the given boundary is a fundamental problem in isogeometric analysis. A good domain parameterization should satisfy the following criteria: (1) the parameterization is a bijective map; and (2) the map has lowest possible distortion. However, none of the state-of-the-art volumetric parameterization methods has fully addressed the above issues. In this paper, we propose a three-stage approach for constructing volumetric parameterization satisfying the above criteria. Firstly, a harmonic map is computed between a unit cube and the computational domain. Then a bijective map modeled by a max-min optimization problem is computed in a coarse-to-fine way, and an algorithm based on divide and conquer strategy is proposed to solve the optimization problem efficiently. Finally, to ensure high quality of the parameterization, the MIPS (Most Isometric Parameterizations) method is adopted to reduce the conformal distortion of the bijective map. We provide several examples to demonstrate the feasibility of our approach and to compare our approach with some state-of-the-art methods. The results show that our algorithm produces bijective parameterization with high quality even for complex domains.

cs.GR↗

arXiv2017

A Low-rank Spline Approximation of Planar Domains

Construction of spline surfaces from given boundary curves is one of the classical problems in computer aided geometric design, which regains much attention in isogeometric analysis in recent years and is called domain parameterization. However, for most of the state-of-the-art parameterization methods, the rank of the spline parameterization is usually large, which results in higher computational cost in solving numerical PDEs. In this paper, we propose a low-rank representation for the spline parameterization of planar domains using low-rank tensor approximation technique, and apply quasi-conformal map as the framework of the spline parameterization. Under given correspondence of boundary curves, a quasi-conformal map with low rank and low distortion between a unit square and the computational domain can be modeled as a non-linear optimization problem. We propose an efficient algorithm to compute the quasi-conformal map by solving two convex optimization problems alternatively. Experimental results show that our approach can produce a bijective and low-rank parametric spline representation of planar domains, which results in better performance than previous approaches in solving numerical PDEs.

cs.CG↗

arXiv2011

The Dimension of Spline Spaces with Highest Order Smoothness over Hierarchical T-meshes

This paper discusses the dimension of spline spaces with highest order smoothness over hierarchical T-meshes over certain type of hierarchical T-meshes. The major step is to set up a bijection between the spline space with highest order smoothness over a hierarchical T-mesh and a univariate spline space whose definition depends on the l-edges of the extended T-mesh. We decompose the univariate spline space into direct sums in the sense of isomorphism using the theory of the short exact sequence in homological algebra. According to the decomposition of the univariate spline space, the dimension formula of the spline space with highest order smoothness over certain type of hierarchical T-mesh is presented. A set of basis functions of the spline space is also constructed.

cs.CG↗