基于简化一阶剪切变形理论的功能梯度扩展等几何材料分布优化

戴钊; 初晨旭; 茆雪明; 汪超

doi:10.21656/1000-0887.460011

基于简化一阶剪切变形理论的功能梯度扩展等几何材料分布优化

doi: 10.21656/1000-0887.460011

戴钊^1,,
初晨旭²,
茆雪明¹,
汪超^2, ,

1.
中交二航局第四工程有限公司，安徽芜湖 241000
2.
安徽工程大学力学重点实验室，安徽芜湖 241000

基金项目:

安徽省住房城乡建设科学计划项目 2023-YF062

国家自然科学基金 52408141

安徽高校协同创新项目 GXXT-2022-082

详细信息

作者简介:
戴钊(1992—)，男，工程师(E-mail: 1950267143@qq.com)

通讯作者:
汪超(1985—)，男，副教授(通信作者. E-mail: wangchao@ahpu.edu.cn)

中图分类号: O341
计量
- 文章访问数: 62
- HTML全文浏览量: 14
- PDF下载量: 14
- 被引次数: 0
出版历程
- 收稿日期: 2025-01-16
- 修回日期: 2025-06-17
- 刊出日期: 2026-05-01

Functionally Graded Extended Isogeometric Material Distribution Optimization Based on the Simple First-Order Shear Deformation Theory

1.
CCCC Second Navigation Bureau Fourth Engineering Co., Wuhu, Anhui 241000, P.R. China
2.
Key Laboratory of Mechanics, Anhui Polytechnic University, Wuhu, Anhui 241000, P.R. China

摘要

摘要: 在实际工程应用中，解决质量优化问题不仅能够有效降低成本，还能显著提升结构性能. 本文针对开孔功能梯度材料的质量优化问题，提出了一种基于简化一阶剪切变形理论(S-FSDT)和扩展等几何分析(XIGA)的求解模型，求解以第一自然频率和屈曲临界参数为约束、质量最小化的优化问题. 优化算法采用基于Lévy飞行改进的人工兔子优化算法(IARO)，显著提升了算法的全局探索能力和摆脱局部最优的能力. 在优化设计中，B样条函数取代了传统的功能梯度材料分布函数，将材料分布的控制点作为设计变量. IARO算法通过CEC’2019测试函数的验证，展现出优越的寻优性能. 算例结果表明了该模型的有效性和可行性，未来可以进一步探索该模型在更复杂工程结构中的应用，实现更全面的结构优化设计.
- 简化一阶剪切变形理论 /
- 扩展等几何 /
- 开孔功能梯度板 /
- 人工兔子优化算法
Abstract: In practical engineering applications, solving the mass optimization problem can not only effectively reduce the cost, but also significantly improve the structural properties. To address this mass optimization problem of functional graded material plates with holes, a solution model was proposed based on the simple first-order shear deformation theory (S-FSDT) and the extended isogeometric analysis (XIGA) to solve the mass minimization problem with the first natural frequency and the buckling critical parameter as the constraints. In the optimization procedure, an improved artificial rabbits optimization (IARO) algorithm incorporating the Lévy flight strategies was employed, which substantially enhances the algorithm's global exploration capability and enables effective escape from local optima. In the optimization design, the B-spline function was used to replace the traditional functionally gradient material distribution function, and the control points of the material distribution were used as design variables. The IARO algorithm demonstrates superior optimization performance through comprehensive benchmark testing through the CEC 2019 test functions, and the results of the arithmetic examples show the validity and feasibility of the proposed model, which can be further explored in the future for the application of the model in more complex engineering structures to achieve a more comprehensive structural optimization design.
- simple first-order shear deformation theory /
- extended isogeometric analysis /
- functionally grade plate with holes /
- artificial rabbit optimization algorithm

HTML全文

图 1 IARO示意图

Figure 1. Schematic of the IARO

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图 2 CEC’2019测试函数收敛图

注为了解释图中的颜色，读者可以参考本文的电子网页版本，后同.

Figure 2. CEC' 2019 test function convergence diagram

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图 3 开心形孔FG方板示意图

Figure 3. A square FGM plate with a heart-shaped cutout

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图 4 开圆孔FG方板示意图

Figure 4. A square FGM plate with a circular hole

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图 5 复杂开孔FG板示意图

Figure 5. The functionally graded plate with a complicated hole

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图 6 复杂开孔FG板的迭代曲线

Figure 6. Iteration curves of complex functionally graded plates with holes

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图 7 最优解的六种自振振型

Figure 7. The six vibration mode shapes of the optimized solutions

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图 8 最优解的六种屈曲振型

Figure 8. The six buckling mode shapes of the optimized solutions

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图 9 双孔FG板示意图

Figure 9. The functionally graded plate with two holes

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图 10 双孔板最优解的六种自振振型

Figure 10. The six vibration mode shapes of the optimized solutions for two-hole plates

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图 11 双孔板最优解的六种屈曲振型

Figure 11. The six buckling mode shapes of the optimized solutions for two-hole plates

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表 1 算法参数设置

Table 1. Algorithm parameter settings

algorithm	parameter setting
NGO	I is 0 or 2
SO	c₁=0.5, c₂=0.05, c₃=2
GWO	a decreasing linearly from 2 to 0
IGWO	a decreasing linearly from 2 to 0
AOA	α=5, μ=0.5

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表 2 不同优化器在CEC’2019基准上的性能统计

Table 2. Performance statistics of different optimizers on CEC'2019 benchmark

function		NGO	SO	WOA	SCA	GWO	IGWO	ARO	IARO
F1	mean	1.10	1.07E6	2.02E6	1.14E6	9.47E2	7.38E4	1.00	1.00
	std	4.06E-1	1.13E6	4.03E6	1.75E6	3.35E3	7.97E4	0.00	0.00
	t	2.13E-1	1.92E-5	1.02E-2	1.28E-3	1.40E-1	2.61E-5	NaN	NaN
	Wilcoxon	+	+	+	+	+	+	≈
F2	mean	9.35	4.90E2	7.14E3	1.79E3	2.18E2	1.02E3	3.91	3.85
	std	4.82	4.43E2	2.36E3	9.84E2	1.70E2	3.67E2	3.23E-1	3.12E-1
	t	1.38E-6	2.07E-6	2.46E-16	7.80E-11	1.98E-7	4.04E-15	5.04E-1	NaN
	Wilcoxon	+	+	+	+	+	+	≈
F3	mean	1.15	3.93	2.95	7.44	1.69	3.42	1.42	1.41
	std	1.94E-1	1.68	1.99	1.59	9.39E-1	1.59	4.06E-2	2.25E-4
	t	9.07E-8	6.62E-9	2.09E-4	6.24E-19	1.22E-1	1.68E-7	3.16E-1	NaN
	Wilcoxon	+	+	+	+	+	+	+
F4	mean	1.46E1	1.66E1	5.38E1	4.16E1	1.39E1	2.20E1	1.49E1	1.39E1
	std	4.09	5.91	1.69E1	7.50	5.61	3.88	5.04	5.46
	t	5.74E-1	8.52E-2	4.17E-12	4.63E-18	9.80E-1	5.26E-7	4.40E-1	NaN
	Wilcoxon	-	-	+	+	+	+	+
F5	mean	1.03	1.66	1.89	6.60	1.64	1.51	1.09	1.08
	std	1.74E-2	7.76E-2	3.75E-1	1.15	6.83E-1	8.53E-2	4.84E-2	4.42E-2
	t	1.92E-6	1.50E-26	1.98E-12	1.11E-21	1.06E-4	1.68E-20	6.69E-1	NaN
	Wilcoxon	+	+	+	+	+	+	+
F6	mean	1.12	4.12	8.39	6.60	2.08	1.00	1.58	1.50
	std	3.34E-1	1.37	1.74	1.15	8.93E-1	5.42E-4	6.37E-1	7.48E-1
	t	2.37E-2	1.66E-10	1.07E-17	1.66E-17	2.06E-2	1.12E-3	6.12E-1	NaN
	Wilcoxon	+	+	+	+	+	-	+
F7	mean	6.01E2	6.38E2	1.21E3	1.34E3	1.75E2	7.51E2	6.22E1	4.53E1
	std	1.58E2	3.42E2	3.00E2	2.05E2	1.40E2	2.42E2	7.48E1	6.55E1
	t	5.57E-17	4.59E-10	2.02E-19	1.15E-23	3.29E-4	1.09E-14	3.18E-1	NaN
	Wilcoxon	+	+	+	+	+	+	+
F8	mean	3.17	3.80	4.31	4.26	3.57	3.22	2.68	2.52
	std	2.90E-1	3.09E-1	4.00E-1	2.33E-1	4.66E-1	3.46E-1	5.21E-1	6.15E-1
	t	2.76E-6	6.12E-11	2.15E-13	8.00E-15	3.29E-7	1.26E-5	2.86E-1	NaN
	Wilcoxon	+	+	+	+	+	+	+
F9	mean	1.13	1.32	1.33	1.43	1.15	1.18	1.15	1.13
	std	3.52E-2	9.36E-2	1.41E-1	1.10E-1	5.96E-2	3.30E-2	7.26E-2	4.88E-2
	t	8.44E-1	3.73E-11	6.62E-8	9.47E-16	1.42E-1	1.46E-4	1.94E-1	NaN
	Wilcoxon	-	+	+	+	+	+	+
F10	mean	1.14E1	2.14E1	2.11E1	2.14E1	2.14E1	2.07E1	1.91E1	1.85E1
	std	9.57	9.93E-2	8.64E-2	5.68E-2	5.81E-2	3.58	5.76	6.39
	t	2.20E-3	2.12E-2	3.82E-2	2.21E-2	2.16E-2	1.32E-1	7.32E-1	NaN
	Wilcoxon	-	+	+	+	+	+	+
	+/-/≈/gm 62/6/2/60

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表 3 第一自然频率对比统计表(单位: rad/s)

Table 3. Comparison of the critical buckling stresses(unit: rad/s)

method/BC	order
method/BC	1	2	3	4
ref. [35]	5.098	6.608	6.929	8.644
ref. [36]	5.193	6.579	6.597	7.819
ref. [37]	4.919	6.398	6.775	8.613
present	4.634	6.856	6.936	8.814

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表 4 临界屈曲应力对比统计表(单位: Pa)

Table 4. Comparison of the critical buckling stresses(unit: Pa)

method	material graded index
method	0	1	5
ref. [38]	6.971	4.686	4.061
ref. [39]	6.992	4.888	4.158
ref. [40]	7.013	4.903	4.170
present	7.009	4.900	4.158

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