Computer Aided Geometric Design最新文献

英文中文

Efficient worst-case topology optimization of self-supporting structures for additive manufacturing 增材制造自支撑结构的高效最坏情况拓扑优化

IF 1.3 4区计算机科学 Q3 COMPUTER SCIENCE, SOFTWARE ENGINEERING

Computer Aided Geometric Design

Pub Date : 2025-04-22 DOI: 10.1016/j.cagd.2025.102441

Nan Zheng, Xiaoya Zhai, Falai Chen

In worst-case topology optimization, uncertain loads can result in complex internal structures and intricate printed details that challenge manufacturability. However, the impact of these features on manufacturing performance is often overlooked, potentially compromising the printability and quality of the final product in additive manufacturing (AM). This paper introduces a novel approach for generating 3D self-supporting structures under worst-case topology optimization. The proposed framework utilizes an implicit tensor-product B-spline (ITPBS) representation, directly adopting its coefficients as design variables to minimize compliance while enforcing self-supporting constraints and minimal length scale. By reformulating AM constraints, we analytically derive a single geometric fabrication constraint that simultaneously addresses both overhang regions and the dripping effect. The solid-void boundary representation provided by ITPBS enables seamless integration of fabrication constraints into the worst-case optimization process. Worst-case compliance is evaluated by solving an eigenvalue problem, and sensitivity analysis is conducted using the adjoint variable method. Numerical experiments demonstrate that the proposed approach effectively produces self-supporting structures across various models.

在最坏情况拓扑优化中，不确定的负载会导致复杂的内部结构和复杂的打印细节，从而挑战可制造性。然而，这些特性对制造性能的影响往往被忽视，可能会影响增材制造（AM）中最终产品的可打印性和质量。介绍了一种基于最坏情况拓扑优化生成三维自支撑结构的新方法。所提出的框架采用隐式张量-积b样条（ITPBS）表示，直接采用其系数作为设计变量，以最小化顺应性，同时强制实施自支持约束和最小长度尺度。通过重新制定增材制造约束，我们解析地推导出一个同时解决悬垂区域和滴水效应的单一几何制造约束。ITPBS提供的固-空边界表示可以将制造约束无缝集成到最坏情况优化过程中。通过求解特征值问题来评估最坏情况下的柔度，并采用伴随变量法进行灵敏度分析。数值实验表明，该方法能有效地生成跨多种模型的自支撑结构。

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引用次数: 0

On the triple-parameter least squares progressive iterative approximation and its convergence analysis 三参数最小二乘渐进迭代逼近及其收敛性分析

IF 1.3 4区计算机科学 Q3 COMPUTER SCIENCE, SOFTWARE ENGINEERING

Computer Aided Geometric Design

Pub Date : 2025-04-17 DOI: 10.1016/j.cagd.2025.102439

Nian-Ci Wu , Chengzhi Liu , Juncheng Li

In the domain of large-scale data fitting, the least squares progressive iterative approximation (LSPIA) method, introduced by Deng and Lin (2014), stands as a prominent figure in geometric iterative techniques due to its simplicity and effectiveness. Herein, we incorporate a triple-parameter modification to fine-tune the control points, hereby denoted as the TLSPIA method. Opting for a suitable parameter within the TLSPIA framework reverts it to the traditional LSPIA and its variant, such as PmLSPIA proposed by Liu et al. (2024). Through singular value decomposition of the collocation matrix, we derive closed forms for the error sequences generated by the LSPIA, PmLSPIA, and TLSPIA methods, demonstrating that these sequences exhibit linear decrease and diverge in their convergence rates across different subspaces corresponding to any right singular vector of the collocation matrix. Empirical numerical examples are provided to substantiate our theoretical claims, highlighting the practical viability and efficacy of the proposed algorithm.

在大规模数据拟合领域，由 Deng 和 Lin（2014 年）提出的最小二乘渐进迭代逼近（LSPIA）方法因其简单有效而成为几何迭代技术中的佼佼者。在此，我们采用三参数修正法对控制点进行微调，即 TLSPIA 方法。在 TLSPIA 框架内选择一个合适的参数，就能将其还原为传统的 LSPIA 及其变体，如 Liu 等人（2024 年）提出的 PmLSPIA。通过对配位矩阵进行奇异值分解，我们推导出了 LSPIA、PmLSPIA 和 TLSPIA 方法产生的误差序列的闭合形式，证明这些序列呈现线性下降，并在与配位矩阵的任意右奇异向量相对应的不同子空间中收敛率发散。我们还提供了经验数值示例来证实我们的理论主张，突出了所提算法的实用性和有效性。

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引用次数: 0

ESA-GS: Elongation splitting and assimilation in Gaussian splatting for accurate surface reconstruction ESA-GS：用于精确表面重建的高斯溅射中的伸长分裂和同化

IF 1.3 4区计算机科学 Q3 COMPUTER SCIENCE, SOFTWARE ENGINEERING

Computer Aided Geometric Design

Pub Date : 2025-04-17 DOI: 10.1016/j.cagd.2025.102434

Yuyang Chen, Wenming Wu, Yusheng Peng, Yue Fei, Liping Zheng

Recently, 3D Gaussian Splatting (3DGS) has significantly advanced the development of 3D reconstruction by providing efficient and high-quality rendering. 2D Gaussian Splatting (2DGS) introduced two-dimensional surfels as scene primitives to address 3DGS's limitations in surface representation. However, its adaptive control strategy may still result in suboptimal results, especially when dealing with extreme-shaped or large Gaussians on the surface. We propose Elongation Splitting and Assimilation in Gaussian Splatting (ESA-GS) to enhance geometric reconstruction quality by addressing these special Gaussians. Specifically, ESA-GS splits highly elongated Gaussians on the surface into three assimilated Gaussians during the densification process. In addition, ESA-GS adds an opacity degeneration strategy and an additional pruning strategy to remove invalid Gaussians and improve the geometry quality. Experimental results demonstrate that ESA-GS can produce geometrically accurate reconstructed surfaces without sacrificing efficiency in most cases.

近年来，三维高斯喷溅技术（3DGS）通过提供高效、高质量的渲染，极大地推动了三维重建的发展。2D高斯喷溅（2DGS）引入二维冲浪作为场景基元，以解决3DGS在表面表示方面的局限性。然而，其自适应控制策略仍然可能导致次优结果，特别是当处理曲面上的极端形状或大高斯分布时。我们提出了高斯溅射（ESA-GS）中的伸长分裂和同化，通过处理这些特殊的高斯分布来提高几何重建质量。具体地说，ESA-GS在致密化过程中将表面上高度拉长的高斯分解为三个同化的高斯。此外，ESA-GS还增加了一种不透明度退化策略和一种额外的剪枝策略，以去除无效高斯分布，提高几何质量。实验结果表明，在大多数情况下，ESA-GS可以在不牺牲效率的情况下产生几何精度的重建表面。

引用次数: 0

Topology guaranteed and error controlled curve tracing for parametric surface-surface intersection 参数曲面相交的拓扑保证和误差控制曲线跟踪

IF 1.3 4区计算机科学 Q3 COMPUTER SCIENCE, SOFTWARE ENGINEERING

Computer Aided Geometric Design

Pub Date : 2025-04-11 DOI: 10.1016/j.cagd.2025.102432

Bingwei Zhang , Jin-San Cheng , Yu-Shen Liu , Zhaoqi Zhang

We present a novel and efficient curve tracing method of the intersection of two parametric surfaces with error bound controlled and correct topology. Our method decomposes the 4D intersection curve in the parameter space into strongly monotonic curve segments such that their corresponding 3D curve segments in the model space are also strongly monotonic. This decomposition strategy can prevent straying or looping and maintain the 3D topology between the curve segments. Furthermore, by controlling the density of decomposition in the model space, our method can easily control the error bound of the numerical approximation of the curve segments. As a result, our method is very efficient compared to previous related work. Our experiments support our claims.

提出了一种误差界可控且拓扑正确的两参数曲面交点曲线跟踪方法。我们的方法将参数空间中的四维相交曲线分解为强单调曲线段，使其在模型空间中对应的三维曲线段也是强单调的。这种分解策略可以防止曲线段之间的偏离或循环，保持曲线段之间的三维拓扑结构。此外，通过控制模型空间中的分解密度，我们的方法可以很容易地控制曲线段数值逼近的误差界。因此，与以往的相关工作相比，我们的方法是非常高效的。我们的实验支持我们的主张。

引用次数: 0

A shift-invariant C12-subdivision algorithm which rotates the lattice 一个平移不变的c12细分算法，它旋转晶格

IF 1.3 4区计算机科学 Q3 COMPUTER SCIENCE, SOFTWARE ENGINEERING

Computer Aided Geometric Design

Pub Date : 2025-03-18 DOI: 10.1016/j.cagd.2025.102430

Cédric Gérot , Malcolm A. Sabin

In order to study differential properties of a subdivision surface at a markpoint, it is necessary to parametrise it over a so-called characteristic map defined as the infinite union of

C^{k}

-parametrised rings. Construction of this map is known when a single step of the subdivision scheme does not rotate a regular lattice. Otherwise, two steps are considered as they realign the lattice and its subdivided version. We present a new subdivision scheme which rotates the lattice and nevertheless allows a direct construction of the characteristic map. It is eigenanalysed with techniques introduced in a companion article and proved to define a

C_{1}^{2}

-algorithm around a face-centre. This scheme generalises Loop's scheme, allowing the designer to choose between extraordinary vertices or faces in regard to the shape of the mesh, the location of the extraordinary elements, and the aimed limit shape.

为了研究标记点处细分曲面的微分性质，有必要在定义为ck参数化环无穷并的特征映射上对细分曲面进行参数化。当细分方案的单个步骤不旋转规则晶格时，该图的构造是已知的。否则，在重新排列晶格及其细分版本时需要考虑两个步骤。我们提出了一种新的细分方案，它旋转晶格，但仍然允许直接构造特征映射。它是特征分析与技术介绍在同伴文章和证明，以定义一个c12算法周围的脸中心。这个方案推广了Loop的方案，允许设计师根据网格的形状、特殊元素的位置和目标极限形状在特殊顶点或面之间进行选择。

引用次数: 0

RBF-MAT: Computing medial axis transform from point clouds by optimizing radial basis functions RBF-MAT：通过优化径向基函数计算点云的中轴变换

IF 1.3 4区计算机科学 Q3 COMPUTER SCIENCE, SOFTWARE ENGINEERING

Computer Aided Geometric Design

Pub Date : 2025-03-04 DOI: 10.1016/j.cagd.2025.102421

Mengyuan Ge , Junfeng Yao , Baorong Yang , Ningna Wang , Zhonggui Chen , Xiaohu Guo

In this paper, we present a simple yet effective method RBF-MAT, for computing medial axis transform (MAT) from point cloud using radial basis functions (RBFs), where the surface is represented as the zero-level set of an interpolating function composed of a linear combination of RBFs. Firstly, we propose a new strategy for selecting the initial medial spheres based on the Voronoi vertices computed from the input points while preserving necessary geometric characteristics. Then the centers and radii of the generated medial spheres are iteratively optimized by minimizing the RBF-based surface reconstruction error. Additionally, the connectivity of the refined medial spheres is constructed as the dual of the restricted power diagram for these spheres. Experimental results across diverse 3D shapes demonstrate our method's efficacy in capturing global structural attributes and local geometric intricacies, with our connectivity approach surpassing existing methods. Besides, the experimental results show that the MATs computed with our method better approximate the point cloud surface than state-of-the-art methods.

在本文中，我们提出了一种简单而有效的方法RBF-MAT，用于使用径向基函数（rbf）计算点云的中轴变换（MAT），其中表面表示为由rbf的线性组合组成的插值函数的零水平集。首先，我们提出了一种基于输入点计算的Voronoi顶点选择初始内侧球体的新策略，同时保留了必要的几何特征。然后，通过最小化基于rbf的曲面重建误差，迭代优化生成的中间球体的中心和半径。此外，精致的中间球体的连通性被构建为这些球体的受限功率图的对偶。不同3D形状的实验结果证明了我们的方法在捕获全局结构属性和局部几何复杂性方面的有效性，我们的连接方法超越了现有方法。此外，实验结果表明，与现有方法相比，本文方法计算的MATs更接近点云表面。

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引用次数: 0

Infinity branches and asymptotic analysis of algebraic space curves: New techniques and applications 代数空间曲线的无穷分支与渐近分析：新技术与应用

IF 1.3 4区计算机科学 Q3 COMPUTER SCIENCE, SOFTWARE ENGINEERING

Computer Aided Geometric Design

Pub Date : 2025-03-01 DOI: 10.1016/j.cagd.2025.102422

Sonia Pérez-Díaz , Li-Yong Shen , Xin-Yu Wang , R. Magdalena-Benedicto

Let

C

represent an irreducible algebraic space curve defined by the real polynomials

f_{i} (x_{1}, x_{2}, x_{3})

for

i = 1, 2

. It is a recognized fact that a birational relationship invariably exists between the points on

C

and those on an associated irreducible plane curve, denoted as

C^{p}

. In this work, we leverage this established relationship to delineate the asymptotic behavior of

C

by examining the asymptotes of

C_{p}

. Building on this foundation, we introduce a novel and practical algorithm designed to efficiently compute the asymptotes of

C

, given that the asymptotes of

C_{p}

have been ascertained.

设C表示一条不可约的代数空间曲线，它由实多项式fi（x1,x2,x3）定义，当i=1,2时。这是一个公认的事实，在C上的点与相关的不可约平面曲线（表示为Cp）上的点之间总是存在着一种两国关系。在这项工作中，我们利用这种已建立的关系，通过检查Cp的渐近线来描述C的渐近行为。在此基础上，我们引入了一种新的实用算法，旨在有效地计算C的渐近线，假设Cp的渐近线已经确定。

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引用次数: 0

Approximation properties over self-similar meshes of curved finite elements and applications to subdivision based isogeometric analysis 曲面有限元自相似网格的近似性质及其在细分等几何分析中的应用

IF 1.3 4区计算机科学 Q3 COMPUTER SCIENCE, SOFTWARE ENGINEERING

Computer Aided Geometric Design

Pub Date : 2025-02-01 DOI: 10.1016/j.cagd.2025.102413

Thomas Takacs

In this study we consider domains that are composed of an infinite sequence of self-similar rings and corresponding finite element spaces over those domains. The rings are parameterized using piecewise polynomial or tensor-product B-spline mappings of degree q over quadrilateral meshes. We then consider finite element discretizations which, over each ring, are mapped, piecewise polynomial functions of degree p. Such domains that are composed of self-similar rings may be created through a subdivision scheme or from a scaled boundary parameterization.

We study approximation properties over such recursively parameterized domains. The main finding is that, for generic isoparametric discretizations (i.e., where

p = q

), the approximation properties always depend only on the degree of polynomials that can be reproduced exactly in the physical domain and not on the degree p of the mapped elements. Especially, in general,

L^{\infty}

-errors converge at most with the rate

h^{2}

, where h is the mesh size, independent of the degree

p = q

. This has implications for subdivision based isogeometric analysis, which we will discuss in this paper.

在本研究中，我们考虑由自相似环的无限序列和相应的有限元空间在这些域上组成的域。使用四边形网格上的分段多项式或张量-积b样条映射来参数化环。然后，我们考虑在每个环上映射的有限元离散化，p次的分段多项式函数。这些由自相似环组成的域可以通过细分方案或缩放边界参数化来创建。我们研究了这种递归参数化域上的近似性质。主要发现是，对于一般的等参离散化（即，p=q），近似性质总是只依赖于可以在物理域中精确再现的多项式的程度，而不依赖于映射元素的程度p。特别是，一般情况下，L∞误差最收敛于速率h2，其中h为网格大小，与度p=q无关。这对基于等几何分析的细分有影响，我们将在本文中讨论。

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引用次数: 0

A C1 simplex-spline basis for the Alfeld split in Rs Rs中Alfeld分裂的C1单样条基

IF 1.3 4区计算机科学 Q3 COMPUTER SCIENCE, SOFTWARE ENGINEERING

Computer Aided Geometric Design

Pub Date : 2025-01-13 DOI: 10.1016/j.cagd.2025.102412

Tom Lyche , Jean-Louis Merrien , Hendrik Speleers

The Alfeld split is obtained by subdividing a simplex in

R^{s}

into

s + 1

subsimplices with the barycenter as one of their vertices. On this split, we consider the space of

C^{1}

splines of degree d (

d \geq s + 1

), for which we construct a basis of simplex-splines with knots at the barycenter and the vertices of the simplex. The basis consists of two types of simplex-splines: firstly Bernstein polynomials with domain points on the facets of the simplex and secondly certain simplex-splines with at least one knot at the barycenter. Partition of unity, Marsden-like identities, and domain points are shown. We also provide

C^{1}

smoothness conditions across a facet between two simplices.

Alfeld分裂是通过将Rs中的一个单纯形细分为s+1个以质心为顶点的子单纯形而得到的。在这个分割上，我们考虑了d次（d≥s+1）的C1样条空间，为此我们构造了一个在质心和单纯形顶点有结点的单纯样条基。该基由两类简单样条组成：一类是在单纯形面上有域点的Bernstein多项式，另一类是在质心处至少有一个结的某些简单样条。给出了单位划分、类马斯登恒等式和定义域点。我们还提供了跨越两个简单面之间的面的C1平滑条件。

引用次数: 0

A novel heterogeneous deformable surface model based on elasticity 基于弹性的新型异质可变形表面模型

IF 1.3 4区计算机科学 Q3 COMPUTER SCIENCE, SOFTWARE ENGINEERING

Computer Aided Geometric Design

Pub Date : 2024-11-13 DOI: 10.1016/j.cagd.2024.102402

Ciyang Zhou, Xingce Wang, Zhongke Wu

The thin membranes and shells in nature are heterogeneous. They are widely used in surgical simulation, biological techniques, and computer animation. The corresponding surface deformable models can implement dynamic simulations of thin membranes and shells in nature, while most surface deformable models are isotropic and cannot represent thin membranes and shells in nature accurately. Therefore, we propose a novel physically-based heterogeneous deformable surface model. By utilizing the same B-spline basis functions or the parameter space of surfaces' geometric representations, we implement material modeling and propose the representations of surfaces with material variations with composite or continuous material functions. Then, we propose a novel physically-based elastic deformable surface model that constructs infinitesimal elements in the parameter space and employs elasticity to analyze their deformation. The corresponding elastic potential energy function is only related to surfaces' continuous representations, and our model avoids the computation error caused by meshes' quality and large rotation of points' frames. We employ isogeometric analysis to solve the dynamic equations derived from our surface model. To demonstrate the validity and reality of our model, several comparison experiments are designed. The corresponding results are in line with expectations and consistent with physical laws.

自然界中的薄膜和外壳是异质的。它们被广泛应用于外科模拟、生物技术和计算机动画。相应的表面可变形模型可以对自然界中的薄膜和薄壳进行动态模拟，而大多数表面可变形模型都是各向同性的，不能准确地表现自然界中的薄膜和薄壳。因此，我们提出了一种基于物理的新型异质可变形表面模型。通过利用相同的 B 样条基函数或曲面几何表示的参数空间，我们实现了材料建模，并提出了用复合材料函数或连续材料函数表示具有材料变化的曲面。然后，我们提出了一种基于物理的新型弹性可变形曲面模型，该模型在参数空间中构建了无穷小元素，并利用弹性分析其变形。相应的弹性势能函数只与曲面的连续表示相关，我们的模型避免了因网格质量和点框架的大幅旋转造成的计算误差。我们采用等距分析法来求解由曲面模型导出的动态方程。为了证明我们模型的有效性和现实性，我们设计了几个对比实验。相应的结果符合预期，并与物理规律相一致。

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