基于规则正交方向场的建筑曲面网壳四边形网格生成方法

doi:10.11996/JG.j.2095-302X.2026051084

摘要/Abstract

摘要：

针对大跨度建筑曲面网壳结构的四边形网格生成需求，提出一种对齐曲面规则正交方向场的网格生成与优化方法。该方法以周期转移函数构建的全局参数化为基础，通过样条曲面的重参数化实现等参线与目标正交方向场的对齐。以样条曲面离散后顶点的重参数坐标为变量，建立度量重参数化梯度与目标方向场偏差的能量泛函模型，并将其离散近似为加权边能量形式，求解能量最小化以得到与方向场对齐的离散重参数域。利用相邻三角形重参数域间的周期转移关系，在重参数域中构建连续分布的参数三角形，提取等参线网格并映射回曲面，生成满足方向约束的初始四边形网格。建立等参线间距与目标网格尺寸之间的映射关系，实现初始四边形网格密度的直接调控，无需重复求解参数化即可快速获得不同尺寸的网格结果。进一步考虑建筑曲面网壳对网格几何的要求，建立包含网格单元平面性、整体顺滑性和尺寸均匀性的多目标优化模型，在曲面原始参数域内对网格顶点位置进行迭代优化，得到兼顾建筑设计要求的网格分布。实验结果表明：该方法能够高效生成对齐任意规则正交方向场的四边形网格，在平面性、均匀性及综合质量指标方面优于对比方法；对初始三角形网格的离散方式具有较强稳健性，并可通过调整重参数域的等参线分布实现网格尺寸的实时控制。网壳结构分析结果进一步表明，在相同杆件截面积和材料参数条件下，优化后的网格分布能够有效降低杆件最大应力并改善结构整体应力分布，而结构用钢量基本保持不变，验证了该方法在建筑曲面网壳设计中的有效性和工程适用性。

关键词: 样条曲面, 正交方向场, 全局重参数化, 四边形网格, 网壳结构

Abstract:

For large-span architectural curved gridshells, a method for generating and optimizing quadrilateral grids aligned with a regular orthogonal direction field on surfaces was proposed. Based on global parameterization constructed using periodic transition functions, the spline surface was reparameterized to align its isoparametric lines with the target orthogonal direction field. Using the reparameterized coordinates of vertices on the discretized surface as variables, an energy functional was formulated to measure the deviation between the gradient of the reparameterization and the direction field. The functional was discretized into a weighted edge-based energy form. Energy minimization yielded a discrete reparameterized domain aligned with the direction field. Based on the periodic transition relationships between adjacent triangular parameter domains, continuous parametric triangles were reconstructed in the reparameterized domain. Isoparametric lines were then extracted and mapped back onto the surface to generate an initial quadrilateral grid satisfying the directional constraints. A mapping between isoparametric line spacing and target grid size was established, enabling direct control of the grid density and rapid generation of grids with different sizes without repeated parameterization. To satisfy the geometric requirements of architectural curved gridshells, a multi-objective optimization model considering grid planarity, smoothness, and size uniformity was further developed. Vertex positions were iteratively optimized in the original parametric domain of the surface to obtain grid layouts compatible with architectural design requirements. Numerical experiments demonstrated that the method efficiently generated quadrilateral grids aligned with arbitrary regular orthogonal direction fields, and outperformed comparison methods in planarity, uniformity, and overall grid quality. The method also exhibited robustness to different initial triangulations and enabled real-time control of grid sizes through adjusting isoparametric lines. Structural analysis of gridshells further showed that, under identical member cross-sections and material properties, the optimized grid layouts effectively reduced the maximum member stress and improved the overall stress distribution, while maintaining almost unchanged steel consumption, demonstrating the effectiveness and engineering applicability of the method for architectural curved gridshell design.

Key words: spline surface, orthogonal direction field, global reparameterization, quadrilateral grid, gridshell structure

中图分类号:

张启涛, 肖骁. 基于规则正交方向场的建筑曲面网壳四边形网格生成方法[J]. 图学学报, 2026, 47(5): 1084-1094.

ZHANG Qitao, XIAO Xiao. Quadrilateral grid generation for architectural curved gridshells guided by regular orthogonal direction field[J]. Journal of Graphics, 2026, 47(5): 1084-1094.

图/表 19

图1 连续重参数域的构造((a) 初始重参数域；(b) 平移后的重参数域)

Fig. 1 Construction of continuous reparameterized domain ((a) Initial reparameterization domain; (b) Reparameterized domain after translation)

图2 重参数域提取网格

Fig. 2 Grid extraction from the reparameterized domain

图3 网格单元平面性和顺滑性表征((a) 相邻边的夹角; (b) 边的方向变化)

Fig. 3 Characterization of grid planarity and smoothness ((a) Angle between adjacent edges; (b) Directional change of edges)

图4 变截面环面网壳四边形网格(重参数化耗时：40 ms)

Fig. 4 Quadrilateral grids of a torus with varying cross sections (reparameterization time: 40 ms)

图5 扭曲环面网壳四边形网格(重参数化耗时：47 ms)

Fig. 5 Quadrilateral grids of a twisted torus (reparameterization time: 47 ms)

图6 花瓣形网壳四边形网格(重参数化耗时：66 ms)

Fig. 6 Quadrilateral grids of a petal-shaped surface (reparameterization time: 66 ms)

图7 不规则环面四边形网格(重参数化耗时：67 ms)

Fig. 7 Quadrilateral grids of an irregular ring surface (reparameterization time: 67 ms)

表1 网格对齐主曲率方向场的量化指标

Table 1 Quantitative metrics for grid alignment with principal curvature direction field

模型	耗时/ms	优化后的能量值(×10^-4)	方向场对齐度/rad			网格正交性/rad
模型	耗时/ms	优化后的能量值(×10^-4)	$\lambda =2.0$	$\lambda =1.0$	$\lambda =0.5$	$\lambda =2.0$	$\lambda =1.0$	$\lambda =0.5$
变截面环面网壳	40	0.653	0.034	0.035	0.035	0.042	0.042	0.023
扭曲环面网壳	47	0.748	0.039	0.028	0.026	0.058	0.042	0.031
花瓣形曲面网壳	66	0.580	0.040	0.042	0.039	0.048	0.047	0.040
不规则环面网壳	67	0.574	0.006	0.009	0.018	0.020	0.015	0.028

图8 含孔洞的曲面四边形网格

Fig. 8 Quadrilateral grids of a surface with a hole

图9 能量泛函优化过程((a) 采用渐变分布的初始三角形网格；(b) 采用随机分布的初始三角形网格)

Fig. 9 Optimization process of the energy functional ((a) Initial triangulation with a gradient distribution; (b) Initially randomly distributed triangulation)

图10 不同权重组合的网格结果((a) 初始四边形网格；(b) 平均权重(${w}_{\text{p}}={w}_{\text{f}}={w}_{\text{d}}={w}_{\text{l}}=0.25$)；(c) 单元平面性(${w}_{\text{p}}=0.7$)；(d) 网格顺滑性(${w}_{\text{f}}=0.7$)；(e) 网格形态变化(${w}_{\text{d}}=0.7$)；(f) 网格尺寸均匀性(${w}_{\text{l}}=0.7$))

Fig. 10 Grid results for different combinations of weights ((a) Initial quadrilateral grids; (b) Average weights (${w}_{\text{p}}={w}_{\text{f}}={w}_{\text{d}}={w}_{\text{l}}=0.25$); (c) Grid planarity (${w}_{\text{p}}=0.7$); (d) Grid smoothness (${w}_{\text{f}}=0.7$); (e) Grid variation (${w}_{\text{d}}=0.7$); (f) Grid size uniformity (${w}_{\text{l}}=0.7$))

表2 不同权重下的网格质量对比

Table 2 Comparison of grid quality under different weights

权重系数	平面性/(rad ×10^-3)	顺滑性/rad	尺寸均匀性
优化前	0.581	0.289	0.301
平均权重	0.129	0.153	0.189
${w}_{\text{p}}=0.7$	0.022	0.235	0.274
${w}_{\text{f}}=0.7$	0.094	0.148	0.284
${w}_{\text{d}}=0.7$	0.054	0.277	0.295
${w}_{\text{l}}=0.7$	0.134	0.270	0.215

图11 螺旋环面

Fig. 11 Helicoidal torus

图12 条带状曲面

Fig. 12 Band surface

图13 不规则曲面

Fig. 13 Irregular surface

表3 不同方法生成的网格质量对比

Table 3 Comparison of grid quality using different methods

模型	平面性/(rad ×10^-3)				顺滑性/rad				尺寸均匀性
模型	MIQ	Instant	Bi-MDF	本文	MIQ	Instant	Bi-MDF	本文	MIQ	Instant	Bi-MDF	本文
螺旋环面	0.061	1.137	0.076	0.027	0.232	0.233	0.198	0.250	0.140	0.137	0.100	0.077
条带状曲面	0.263	0.265	0.240	0.230	0.094	0.089	0.098	0.081	0.104	0.061	0.097	0.108
不规则曲面	0.165	0.372	0.229	0.153	0.158	0.184	0.113	0.123	0.121	0.068	0.494	0.108

图14 网壳网格优化前后的网格质量云图(左：优化前；右：优化后) ((a) 网格形态；(b) 单元平面性/(rad ×10-3)；(c) 网格顺滑性；(d) 网格尺寸均匀性)

Fig. 14 Grid quality before (left) and after (right) shape optimization ((a) Grid shape; (b) Grid planarity/(rad ×10-3); (c) Grid smoothness; (d) Grid size uniformity)

图15 最不利工况下的结构组合应力云图

Fig. 15 Structural combined stress results under the most unfavorable load condition

图16 荷载标准组合下的结构组合位移云图

Fig. 16 Structural combined displacement results under the standard load condition

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