Quadrilateral grid generation for architectural curved gridshells guided by regular orthogonal direction field

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

Abstract

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

CLC Number:

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.

Figures/Tables 19

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

Fig. 2 Grid extraction from the reparameterized domain

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

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

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

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

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

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

Fig. 8 Quadrilateral grids of a surface with a hole

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

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 (${w}_{\text{l}}=0.7$))

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

Fig. 11 Helicoidal torus

Fig. 12 Band surface

Fig. 13 Irregular surface

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

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)

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

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

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