基于流形装配的流形/非流形体统一表达技术研究

doi:10.11996/JG.j.2095-302X.2026030629

摘要/Abstract

摘要：

针对已有内核在形体表示范围上的不足，以及复杂形体建模的效率瓶颈问题，提出一种基于流形装配的流形/非流形体统一表达技术，通过将非流形体解析为有限流形体的并集，突破了传统方法必须借助非流形对象或非流形限定关系才能表达非流形体的局限，在不损失形体表达范围的前提下简化了形体数据结构的构建逻辑，为改善流形/非流形体造型算法的效率和通用性奠定基础。此外，还提出一种混合式拓扑关系更新机制，在保证拓扑一致性的前提下将复杂模型(含有10⁴量级拓扑对象的模型)的拓扑检索效率较OCC提升近90%，接近CGM等国际先进商业内核水平，为进一步提升系统建模效率提供了技术支撑。

关键词: 非流形, 流形装配, 形体表达, 拓扑结构, 拓扑链接

Abstract:

To address the limitations of current kernels in solid representation and the efficiency bottlenecks in complex solid modeling, a manifold/non-manifold unified solid representation method based on the manifold assembly concept was proposed. By interpreting non-manifold solids as a union of finite manifold subsets, this approach overcome the limitation of traditional approaches, which relied on non-manifold entities or non-manifold constraints to represent non-manifold solids. The construction logic of the solid representation data structure was simplified without compromising the range of solid representations. This laid the foundation for improving the efficiency and versatility of manifold/non-manifold modeling algorithms. Furthermore, a bidirectional and flexible topological linking mechanism, along with a hybrid topological-link maintenance strategy, was proposed. While ensuring topological consistency, the proposed method improved the topological retrieval efficiency of complex models (containing topological entities at the 10⁴ level) by nearly 90% compared with OCC, approaching the performance of leading international commercial kernels such as CGM. This provided technical support for further enhancing system modeling efficiency.

Key words: non-manifold, manifold assembly, shape representation, topological structure, topological link

中图分类号:

TP391.72

徐保文, 陈辉, 李吉星, 张博文, 蓝健文. 基于流形装配的流形/非流形体统一表达技术研究[J]. 图学学报, 2026, 47(3): 629-640.

XU Baowen, CHEN Hui, LI Jixing, ZHANG Bowen, LAN Jianwen. Research on unified representation technology for manifold/non-manifold shapes based on manifold assembly[J]. Journal of Graphics, 2026, 47(3): 629-640.

图/表 19

图1 形体表达数据结构发展脉络((a) 翼边结构; (b) 辐射边结构; (c) 混合边结构; (d) S集表示法; (e) SGC结构; (f) 单纯复形结构)

Fig. 1 The developmental trajectory of data structures for shape representation ((a) Winged-edge structure; (b) Radial-edge structure; (c) Hybrid-edge structure; (d) S-set representation; (e) Selective geometric complex; (f) Simplicial complex)

图2 形体表达数据结构的表示域范围对比

Fig. 2 A comparative analysis of the representational domain scope of data structures for shape representation

图3 基于复形理论的非流形建模拓扑结构[10]

Fig. 3 Non-manifold modeling topological structure based on complex theory [10]

图4 Edge, Vertex, CoEdge和CoVertex关联关系示意图

Fig. 4 The relationship diagram of Edge, Vertex, CoEdge and CoVertex

图5 构建不同维度Body的分解示意图((a) 0维度; (b) 1维度; (c) 2维度; (d) 3维度)

Fig. 5 Schematic illustration of the decomposition process for bodies across different dimensions ((a) 0-dimension; (b) 1-dimension; (c) 2-dimension; (d) 3-dimension)

图6 已有方法采用的2种非流形体表示方法((a) 引入非流形关系; (b) 引入非流形拓扑元素)

Fig. 6 Two Non-Manifold shape representation methods in existing approaches ((a) By introducing non-manifold relationships; (b) By Introducing non-manifold topological elements)

图7 本文方法表示非流形体示意图

Fig. 7 Schematic illustration of a non-manifold shape represented by the proposed method

图8 拓扑对象之间的限定关系

Fig. 8 The limitation relationships between topological objects

图9 基于流形装配的拓扑结构形体表达示意图

Fig. 9 Topological structure shape representation diagram based on manifold assembly

图10 三类典型非流形体((a) 低维连通; (b) 边界复用; (c) 混合维度)[25]

Fig. 10 Three types of non-manifold shapes ((a) Low- dimensional connectivity; (b) Boundary reuse; (c) Mixed dimensions)[25]

图11 复杂非流形体的2种表示方法((a) 已有方法; (b) 本文方法)

Fig. 11 Two representation methods for complex non- manifold shapes ((a) Existing method; (b) Ours)

图12 拓扑环方向决定CoEdge的存储顺序

Fig. 12 The direction of the Loop determines the storage order of CoEdges

图13 CoEdge与Loop的相对方向

Fig. 13 Relative direction of CoEdge and Loop

图14 各内核拓扑关系存储方案示意图((a) OCC拓扑关系; (b) 本文拓扑关系)

Fig. 14 Topological relationship storage schemes in various kernels diagram ((a) Topological relationship; (b) Topological relationship of this paper)

图15 局部更新、全局更新示意图((a) 局部更新; (b) 全局更新)

Fig. 15 Local and global update diagram ((a) Local update; (b) Global update)

图16 本文方法构建的非流形形体示意图

Fig. 16 Diagram of the non-manifold body constructed by the method in this paper

图17 测试模型汇总((a) 导筒安装座模型; (b) 某环形零件模型; (c) 某梁体零件模型; (d) 某汽缸盖零件模型; (e) 某壁板零件模型; (f) 某动力装置模型; (g) 某机壳零件模型; (h) 直升机组装模型)

Fig. 17 Summary of test models ((a) Guide tube mounting seat model; (b) Annular part model; (c) Beam structure model; (d) Cylinder head model; (e) Wall panel model; (f) Powerplant assembly model; (g) Housing/Casing model; (h) Helicopter assembly model)

表1 向下拓扑链接效率测试结果/ms

Table 1 Test results of downward topological link efficiency/ms

测试模型/(拓扑对象数量级)	CGM	OCC	ACIS	本文方法
直升机组装模型/(3×10⁴)	7.84	72.74	27.32	16.77
导筒安装座模型/(7×10³)	1.85	18.13	5.91	2.35
某环形零件模型/(4×10³)	0.97	10.20	3.94	1.32
某汽缸盖零件模型/(2×10³)	0.42	8.67	2.40	0.63
某对称件零件模型/(2×10³)	0.39	8.62	1.93	0.57
某动力装置模型/(2×10³)	0.52	8.24	2.17	0.55
某梁体零件模型/(2×10³)	0.37	7.90	2.53	0.34
某机壳零件模型/(1×10³)	0.23	6.01	1.40	0.21

表2 向上拓扑链接效率测试结果/ms

Table 2 Test results of upward topological link efficiency/ms

测试模型/(拓扑对象数量级)	CGM	OCC	ACIS	本文方法
直升机组装模型/(3×10⁴)	15.61	8.25×10⁶	98.68	56.23
导筒安装座模型/(7×10³)	4.23	4.80×10⁵	18.22	10.68
某环形零件模型/(4×10³)	2.47	1.57×10⁵	11.74	5.27
某汽缸盖零件模型/(2×10³)	1.74	6.07×10⁴	7.51	2.71
某对称件零件模型/(2×10³)	1.12	4.51×10⁴	6.64	2.87
某动力装置模型/(2×10³)	1.36	4.64×10⁴	5.26	2.44
某梁体零件模型/(2×10³)	0.76	2.82×10⁴	8.33	1.92
某机壳零件模型/(1×10³)	0.68	2.05×10⁴	5.34	1.66

参考文献 26

[1]	BRAID I C. Designing with volumes[D]. Cambridge: University of Cambridge, 1973.
[2]	BAUMGART B G. Winged edge polyhedron representation[R]. Stanford: Stanford University, 1972.
[3]	EASTMAN C M. Introduction to computer aided design[R]. Pittsburg: Carnegie-Mellon University, 1982.
[4]	MANTYLA M. A note on the modeling space of Euler operators[J]. Computer Vision, Graphics, and Image Processing, 1984, 26(1): 45-60. DOI URL
[5]	MANTYLA N, SULONEN N. GWB: a solid modeler with Euler operators[J]. IEEE Computer Graphics and Applications, 1982, 2(7): 17-31.
[6]	WEILER K J. Topological structures for geometric modeling[D]. Troy: Rensselelaer Polytechnic Institute, 1986.
[7]	KALAY Y E. The hybrid edge: a topological data structure for vertically integrated geometric modelling[J]. Computer-Aided Design, 1989, 21(3): 130-140. DOI URL
[8]	ARBAB F. Set models and Boolean operations for solids and assemblies[J]. IEEE Computer Graphics and Applications, 1990, 10(6): 76-86.
[9]	ROSSIGNAC J R, O’CONNOR M A. SGC: a dimension- independent model for pointsets with internal structures and incomplete boundaries[M]. Yorktown Heights: IBM TJ Watson Research Center, 1989.
[10]	MASUDA H. Topological operators and Boolean operations for complex-based nonmanifold geometric models[J]. Computer-Aided Design, 1993, 25(2): 119-129. DOI URL
[11]	PAOLUZZI A, BERNARDINI F, CATTANI C, et al. Dimension-independent modeling with simplicial complexes[J]. ACM Transactions on Graphics, 1993, 12(1): 56-102. DOI URL
[12]	武仲科, 戴国忠. 非流形几何模型的拓扑表示——粘合边结构[J]. 航空学报, 1997, 18(4): 432-438.
	WU Z K, DAI G Z. Representation of non manifold——the identification edge structure[J]. Acta Aeronautica et Astronautica Sinica, 1997, 18(4): 432-438(in Chinese).
[13]	邹强. 浅谈实体建模: 历史、现状与未来[J]. 图学学报, 2022, 43(6): 987-1001.
	ZOU Q. A note on solid modeling: history, state of the art, future[J]. Journal of Graphics, 2022, 43(6): 987-1001(in Chinese).
[14]	QI Charles R, SU H, KAICHUN M, et al. PointNet: deep learning on point sets for 3D classification and segmentation[C]// IEEE Conference on Computer Vision and Pattern Recognition. New York: IEEE Press, 2017: 77-85.
[15]	申则宇, 王文成, 侯飞, 等. 加强形状感知的CAD网格模型分割方法[J]. 计算机辅助设计与图形学学报, 2025, 37(11): 1898-1908.
	SHEN Z Y, WANG W C, HOU F, et al. Mesh segmentation with enhanced shape perception for CAD models[J]. Journal of Computer-Aided Design & Computer Graphics, 2025, 37(11): 1898-1908(in Chinese).
[16]	曹杰, 关振群, 李震波. 基于多重网格尺寸场的非结构自适应网格生成[J/OL]. 计算机辅助设计与图形学学报, (2025-07-25) [2025-07-29]. https://link.cnki.net/urlid/11.2925.TP.20250724.1845.022.
	CAO J, GUAN Z Q, LI Z B. Unstructured adaptive mesh generation based on the multi-grid size field[J/OL]. Journal of Computer-Aided Design & Computer Graphics, (2025-07-25) [2025-07-29]. https://link.cnki.net/urlid/11.2925.TP.20250724.1845.022. (in Chinese).
[17]	胡文恺, 马鸿宇, 刘亚醉, 等. T样条用于计算机辅助设计、分析和制造的新型表示方法[J]. 图学学报, 2022, 43(6): 1018-1033.
	HU W K, MA H Y, LIU Y Z, et al. T-splines a new representation for CAD, CAE and CAM[J]. Journal of Graphics, 2022, 43(6): 1018-1033(in Chinese). DOI
[18]	李军成, 刘成志, 刘珊君, 等. 带局部参数的五次Catmull-Rom样条[J]. 计算机辅助设计与图形学学报, 2025, 37(11): 1858-1868.
	LI J C, LIU C Z, LIU S J, et al. The quintic Catmull-Rom spline with local parameters[J]. Journal of Computer-Aided Design & Computer Graphics, 2025, 37(11): 1858-1868(in Chinese).
[19]	太志伟, 张兴龙, 尹晓龙, 等. 三维船舶模型边界表示转构造表示方法[J]. 计算机辅助设计与图形学学报, 2023, 35(12): 1851-1862.
	TAI Z W, ZHANG X L, YIN X L, et al. A method for transforming B-Rep of 3D ship models into CSG[J]. Journal of Computer-Aided Design & Computer Graphics, 2023, 35(12): 1851-1862(in Chinese).
[20]	尹晓龙, 沈恩亚, 王建民. 典型三维模型表示方法转换技术综述[J]. 计算机辅助设计与图形学学报, 2025, 37(10): 1666-1680.
	YIN X L, SHEN E Y, WANG J M. A review of conversion techniques for typical 3D model representation methods[J]. Journal of Computer-Aided Design & Computer Graphics, 2025, 37(10): 1666-1680(in Chinese).
[21]	胡丁允, 张淦宣, 陈彦如, 等. 基于约束最小二乘N边洞NURBS曲面填充算法[J]. 计算机辅助设计与图形学学报, 2025, 37(10): 1717-1725.
	HU D Y, ZHANG G X, CHEN Y R, et al. A Filling algorithm for N-sided holes based on constrained least square method[J]. Journal of Computer-Aided Design & Computer Graphics, 2025, 37(10): 1717-1725(in Chinese).
[22]	胡新宇, 陈鑫, 赵启轩, 等. 带引导线的扩展Coons曲面算法[J]. 计算机辅助设计与图形学学报, 2025, 37(10): 1696-1704.
	HU X Y, CHEN X, ZHAO Q X, et al. An algorithm to generate extended coons surface with guide curves[J]. Journal of Computer-Aided Design & Computer Graphics, 2025, 37(10): 1696-1704(in Chinese).
[23]	DE FLORIANI L, HUI A, PANOZZO D, et al. A dimension-independent data structure for simplicial complexes[M]// SHONTZ S. Proceedings of the 19th International Meshing Roundtable. Heidelberg: Springer, 2010: 403-420.
[24]	LÉON J C, DE FLORIANI L, HÉTROY F. Classification of non-manifold singularities from transformations of 2-manifolds[C]// 2009 IEEE International Conference on Shape Modeling and Applications. New York: IEEE Press, 2009: 179-184.
[25]	XU B W, ZHANG B W, LI Z Z. Geometric modeling techniques based on simplicial complexes[C]// 2024 10th Asia Conference on Mechanical Engineering and Aerospace Engineering. New York: IEEE Press, 2024: 1349-1356.
[26]	孙家广, 陈玉健. 几何造型中的数据结构[J]. 计算机学报, 1989(3): 181-193.
	SUN J G, CHEN Y J. Data structure for geometric modelling[J]. Chinese Journal of Computers, 1989(3): 181-193(in Chinese).