Buckling Topological Design of Reinforced Plate/Shell Based on Bionic Diffusion Growth-driven Method

doi:10.3969/j.issn.1004-132X.2026.02.020

Abstract

Abstract:

To overcome the limitations of insufficient design spaces in existing Bio-inspired reinforced-plate design（BRD） methods resulting from the modeling strategy of the ground structure method （GSM）， a diffusion-based growth scheme inspired by the vein growth patterns of plants was proposed. The scheme simulated the formation processes of plant venation to identify efficient stiffener layouts that enhanced the buckling load-bearing capacity of the structure. An improved optimization modeling framework was adopted， using S9R5 Shell elements and B32/B31 beam elements to simulate the plate（leaf） and stiffeners（veins） respectively. A scheme of node pre-storage and reserve/updating employed to expand the flexibility of stiffeners by increasing the updating-range of active nodes. Rectangular thin plates were investigated as representative examples， buckling analyses under various parameters， boundary conditions（SSSS and SFSF）， and loading cases were conducted to validate the effectiveness of the proposed method. The numerical results demonstrate that the diffusion-based growth scheme yields more efficient and clearer stiffener layouts compared to existing growth strategies.

Key words: reinforced shell, diffusion-driven growth, structure topology, linear buckling analysis, ground structure method

摘要：

针对现有加筋板仿生设计存在因基结构建模方式所引起的优化空间不足的问题，根据植物筋脉生长特征，提出一种扩散式生长方案来模拟植物筋脉的形成过程，寻求加筋的合理布局，使得结构具备高效的屈曲承载能力。采用一种改进的优化建模框架，使用S9R5单元以及B32/B31梁单元模拟叶片和筋脉，并使用节点预存储、保留更新的方案，通过提高生长节点更新范围的方式扩展自适应加筋的灵活性。以矩形薄板为例，通过对不同参数下简支板在四边简支（SSSS）以及双侧简支（SFSF）条件下不同受载模式的屈曲案例进行对比，验证了算法的有效性。结果表明，扩散式生长加筋方案所得到的加筋布局比现有生长方式更为有效，布局清晰。

关键词: 加筋薄壳, 扩散式生长, 结构拓扑, 线性屈曲分析, 基结构法

CLC Number:

TB472

JIANG Xuetao, YANG Yong, ZHU Jihong, FENG Hao, PAN Shunyang. Buckling Topological Design of Reinforced Plate/Shell Based on Bionic Diffusion Growth-driven Method[J]. China Mechanical Engineering, 2026, 37(2): 452-465.

姜学涛, 杨勇, 朱季红, 冯昊, 潘顺洋. 基于仿生扩散生长驱动的加筋板壳抗屈曲拓扑设计[J]. 中国机械工程, 2026, 37(2): 452-465.

Figures/Tables 21

Fig.1 Process of branch

Fig.2 Modeling scheme of ground structure

Fig.4 Schematic diagram of diffusion growth-driven reinforcement

Fig.6 Design flowchart

Fig.7 Design model of square simply supported plate（SSSS）

Fig.8 Case 1（a）：grow from 4 corners（ΔV¯=2.5%）

Fig.9 Case 1（b）：grow from the midpoints of the 4 sides of the square plate（ΔV¯=2.5%）

Tab.1 Numerical results of square plate simply supported on 4 edges（SSSS，κ=25%，ΔV¯=2.5%）

Items	k	体积（ $Δ V k$ （ $%$ ））	屈曲载荷因子（ $λ 1 k$ ， $Δ λ 1 k$ （ $%$ ））
初始	0	0.00	1.5910	0.00
Case 1（a）	38	+2.54	2.5788	+62.09
Case 1（b）	39	+2.54	2.4583	+54.51
Case 1（c）	35	+2.51	2.5506	+60.31
Case 1（d）	40	+2.49	2.7857	+75.09

Tab.1 Numerical results of square plate simply supported on 4 edges（SSSS，κ=25%，ΔV¯=2.5%）

Items	k	体积（ $Δ V k$ （ $%$ ））	屈曲载荷因子（ $λ 1 k$ ， $Δ λ 1 k$ （ $%$ ））
初始	0	0.00	1.5910	0.00
Case 1（a）	38	+2.54	2.5788	+62.09
Case 1（b）	39	+2.54	2.4583	+54.51
Case 1（c）	35	+2.51	2.5506	+60.31
Case 1（d）	40	+2.49	2.7857	+75.09

Fig.10 Case 1（c）：grow from the central region of the square plate（ΔV¯=2.5%）

Fig.11 Case 1（d） grow from 9 nodes distributed in the square plate（ΔV¯=2.5%）

Fig.12 Iterative process of the first buckling mode of square plate under SSSS BC condition（SSSS，ΔV¯=2.5%）

Fig.13 Design model of rectangle plate with simply supported BC condition （SSSS）

Fig.14 Design result of rectangle plate with simply supported on 4 edges under four-sided axial pressure（a∶b=4∶3 with SSSS BC condition， ΔV¯=5%）

Fig.15 Design result of rectangle plate with simply supported on 4 edges under unilateral axial pressure（a∶b=4∶3 with SSSS BC condition， ΔV¯=5%）

Tab. 2 Numerical results of rectangle plate with simply supported on 4 edges（a∶b=4∶3， ΔV¯=5%）

	k	Case 2				k	Case 3
	k	体积（ $Δ V k$ （ $%$ ））	屈曲因子（ $λ 1 k$ ， $Δ λ 1 k$ （ $%$ ））			k	体积（ $Δ V k$ （ $%$ ））	屈曲因子（ $λ 1 k$ ， $Δ λ 1 k$ （ $%$ ））
初始	0	0.00	3.3160	0.00	初始	0	0.00	9.0370	0.00
图17b	/	+5.22	3.8876	+17.24	图17c	/	+5.22	10.5650	+16.91
图14a $κ = 25 %$ $N S e e d s = 10$	15	+0.91	3.8123	+14.97	图15a $κ = 25 %$ $N S e e d s = 10$	5	+0.37	9.70220	+7.36
	25	+1.53	4.0328	+21.60		15	+1.10	10.3470	+14.50
	50	+4.43	4.5845	+38.25		35	+3.21	11.8090	+30.67
	62	+5.17	4.9271	+48.59		47	+5.13	13.2090	+46.17
图14b $κ = 15 %$ $N S e e d s = 10$	25	+1.46	4.0062	+20.81	图15b $κ = 15 %$ $N S e e d s = 10$	5	+0.37	9.7022	+7.36
	50	+2.84	4.4178	+33.23		15	+0.97	10.2790	+13.74
	75	+4.15	4.8734	+46.97		35	+3.77	11.9580	+33.32
	90	+5.06	5.5211	+57.45		45	+5.08	13.0540	+44.45
图14c $κ = 15 %$ $N S e e d s = 12$	25	+1.54	4.0067	+20.83	图15c $κ = 15 %$ $N S e e d s = 12$	5	+0.37	9.7010	+7.35
	50	+2.94	4.4512	+34.23		15	+1.14	10.3630	+14.67
	75	+4.29	4.8240	+45.48		35	+3.67	12.2990	+36.10
	87	+5.02	5.0161	+51.27		45	+5.09	13.3050	+47.23

Tab. 2 Numerical results of rectangle plate with simply supported on 4 edges（a∶b=4∶3， ΔV¯=5%）

	k	Case 2				k	Case 3
	k	体积（ $Δ V k$ （ $%$ ））	屈曲因子（ $λ 1 k$ ， $Δ λ 1 k$ （ $%$ ））			k	体积（ $Δ V k$ （ $%$ ））	屈曲因子（ $λ 1 k$ ， $Δ λ 1 k$ （ $%$ ））
初始	0	0.00	3.3160	0.00	初始	0	0.00	9.0370	0.00
图17b	/	+5.22	3.8876	+17.24	图17c	/	+5.22	10.5650	+16.91
图14a $κ = 25 %$ $N S e e d s = 10$	15	+0.91	3.8123	+14.97	图15a $κ = 25 %$ $N S e e d s = 10$	5	+0.37	9.70220	+7.36
	25	+1.53	4.0328	+21.60		15	+1.10	10.3470	+14.50
	50	+4.43	4.5845	+38.25		35	+3.21	11.8090	+30.67
	62	+5.17	4.9271	+48.59		47	+5.13	13.2090	+46.17
图14b $κ = 15 %$ $N S e e d s = 10$	25	+1.46	4.0062	+20.81	图15b $κ = 15 %$ $N S e e d s = 10$	5	+0.37	9.7022	+7.36
	50	+2.84	4.4178	+33.23		15	+0.97	10.2790	+13.74
	75	+4.15	4.8734	+46.97		35	+3.77	11.9580	+33.32
	90	+5.06	5.5211	+57.45		45	+5.08	13.0540	+44.45
图14c $κ = 15 %$ $N S e e d s = 12$	25	+1.54	4.0067	+20.83	图15c $κ = 15 %$ $N S e e d s = 12$	5	+0.37	9.7010	+7.35
	50	+2.94	4.4512	+34.23		15	+1.14	10.3630	+14.67
	75	+4.29	4.8240	+45.48		35	+3.67	12.2990	+36.10
	87	+5.02	5.0161	+51.27		45	+5.09	13.3050	+47.23

Fig.16 Iterative process of the first buckling mode of rectangle plate with simply supported on 4 edges （a∶b=4∶3，ΔV¯=5%）

Fig.17 Case of orthogonal grid stiffened rectangle plate（a∶b=4∶3，ΔV¯=5%）

Fig.18 Example of a rectangular plate with bilateral simply supported（SFSF） and unilateral axial compression

Fig.19 Design result of rectangle plate with simply supported on ane edge under unilateral axial pressure（a∶b=2∶1 with SFSF BC condition）

Tab.3 Result of rectangle plate with simply supported on ane edge under unilateral axial pressure （a∶b=2∶1 with SFSF BC condition）

	k	体积（ $V k$ （ $%$ ））	屈曲因子（ $λ 1 k$ ， $Δ λ 1 k$ （ $%$ ））
初值	0	0.00	0.4848	0.00
图19 Case 4 $κ = 15 %$ $N S e e d s = 8$	25	+0.79	0.5854	+20.76
	50	+1.44	0.6491	+33.88
	75	+2.09	0.6889	+42.10
	100	+3.12	0.8069	+66.44
	108	+3.51	0.8114	+67.36

Tab.3 Result of rectangle plate with simply supported on ane edge under unilateral axial pressure （a∶b=2∶1 with SFSF BC condition）

	k	体积（ $V k$ （ $%$ ））	屈曲因子（ $λ 1 k$ ， $Δ λ 1 k$ （ $%$ ））
初值	0	0.00	0.4848	0.00
图19 Case 4 $κ = 15 %$ $N S e e d s = 8$	25	+0.79	0.5854	+20.76
	50	+1.44	0.6491	+33.88
	75	+2.09	0.6889	+42.10
	100	+3.12	0.8069	+66.44
	108	+3.51	0.8114	+67.36

Fig.20 Iterative process with CFSF BC condition

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