基于响应面法和改进算术优化算法的抱杆优化设计

陶然; 周焕林; 孟增; 杨小猛

doi:10.21656/1000-0887.420318

基于响应面法和改进算术优化算法的抱杆优化设计

doi: 10.21656/1000-0887.420318

合肥工业大学土木与水利工程学院，合肥 230009

基金项目: 国家自然科学基金（11672098）

详细信息

作者简介:
陶然（1991—），男，博士生（E-mail: taoran127@mail.hfut.edu.cn）

周焕林（1973—），男，教授，博士，博士生导师（通讯作者. E-mail: zhouhl@hfut.edu.cn）

孟增（1986—），男，副教授，博士，博士生导师（通讯作者. E-mail: mengz@hfut.edu.cn）

中图分类号: TM752
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出版历程
- 收稿日期: 2021-10-25
- 修回日期: 2022-05-25
- 网络出版日期: 2022-05-26
- 刊出日期: 2022-10-31

Optimization Design of Holding Poles Based on the Response Surface Methodology and the Improved Arithmetic Optimization Algorithm

College of Civil Engineering, Hefei University of Technology, Hefei 230009, P.R.China

摘要

摘要:
抱杆优化设计需要耗费大量有限元分析计算时间，难以确定可行域。该文采用响应面法（response surface method，RSM）来模拟抱杆结构的真实响应，提出了改进的算术优化算法（improved arithmetic optimization algorithm，IAOA）对抱杆结构进行优化设计。将分数阶积分引入算术优化算法（arithmetic optimization algorithm，AOA），改善了算法的开发能力。采用拉丁超立方抽样，选取抱杆结构杆件截面试验样本，利用最小二乘法对样本点进行分析，构建了抱杆结构应力和位移关于杆件截面尺寸的二阶响应面代理模型。建立以抱杆质量最小化为优化目标，许用应力和位移为约束条件的优化模型，采用IAOA对其进行求解。结果表明：二阶响应面模型能够准确预测抱杆结构的响应值，IAOA的求解精度得到显著提升，代理模型可大幅降低有限元分析所需的计算代价，优化后抱杆结构质量减轻了8.2%。联合使用RSM和IAOA可有效求解大型空间杆系结构的优化设计问题。
- 响应面法 /
- 算术优化算法 /
- 分数阶积分 /
- 代理模型 /
- 抱杆
Abstract:
The computation consumption of finite element analysis for structural optimization design of holding poles is large, and it is difficult to determine the feasible region. The response surface method (RSM) was used to simulate the real response of the holding pole, and an improved arithmetic optimization algorithm (IAOA) was proposed to optimize the holding pole. The fractional-order calculus was introduced into the arithmetic optimization algorithm (AOA) to improve the exploitation ability of the AOA. The Latin hypercube sampling was applied to select the test samples of each member of the holding pole, and the least square method was employed to analyze the sample points. Then, the 2nd-order response surface surrogate model for the stress and displacement of the holding pole on the cross-sectional sizes of each member was established. An optimization model was constructed with the minimum mass as the optimization objective and the allowable stress and displacement as constraints, and the IAOA was implemented to solve the model. The results show that, the 2nd-order response surface model can accurately predict the response value of the holding pole. The solution accuracy of the IAOA is significantly improved. The surrogate model can greatly decrease the calculation cost of the finite element analysis. The mass of the holding pole is reduced by 8.2% after optimization. The RSM and the IAOA can be combined to solve the optimization design problem of large spatial truss structures effectively.
- response surface method /
- arithmetic optimization algorithm /
- fractional-order calculus /
- surrogate model /
- holding pole

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图 1 AOA优化流程

Figure 1. The optimization process of AOA

下载: 全尺寸图片幻灯片

图 2 抱杆

Figure 2. The holding pole

下载: 全尺寸图片幻灯片

图 3 不同算法收敛曲线对比

Figure 3. Comparison of convergence curves for different algorithms

下载: 全尺寸图片幻灯片

表 1 抱杆结构拉丁超立方试验设计结果

Table 1. Latin hypercube experimental design results of the holding pole

number	b₁/mm	t₁/mm	b₂/mm	t₂/mm	b₃/mm	t₃/mm	σ/MPa	d/mm
1	112	10	41	3	81	7	112.776	618.502
2	96	8	46	8	73	6	144.207	855.603
3	145	7	87	7	76	4	112.785	640.648
4	147	8	56	4	96	7	112.759	589.670
5	131	11	94	4	44	8	112.745	503.594
6	91	8	85	4	49	5	154.753	879.046
7	143	10	86	6	58	4	112.738	496.787
8	90	8	77	8	81	7	157.628	931.646
9	127	9	65	6	87	7	112.765	618.600
10	109	8	93	6	56	5	120.008	741.433
11	104	10	49	9	60	8	112.789	672.708
12	115	6	47	6	84	8	158.176	925.155
13	140	12	55	8	67	4	112.725	455.399
14	110	9	51	3	60	5	112.800	672.017
15	137	9	92	4	79	5	112.755	563.238
16	128	10	63	5	85	8	112.750	571.317
17	94	8	84	7	80	4	147.380	865.021
18	141	12	64	6	86	8	112.726	472.743
19	121	9	52	4	87	5	112.793	623.275
20	123	12	72	6	54	4	112.746	498.627
21	120	7	76	8	99	7	123.901	800.895
22	106	7	54	7	88	9	147.233	893.174
23	97	11	56	4	94	6	112.787	667.220
24	100	7	76	4	68	5	158.786	903.924
25	147	11	45	4	89	6	112.729	467.172
26	119	10	50	6	50	9	112.764	590.889
27	129	11	69	5	49	3	112.746	507.854
28	105	7	88	4	95	9	148.844	902.907
29	144	11	98	5	91	5	112.723	480.494
30	127	10	87	5	52	8	112.753	558.483

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表 2 响应面模型随机样本点检验结果

Table 2. Test results of random sample points for the response surface model

item	δ²	δ_RAAE	δ_RMAE
σ/MPa	0.9257	0.0115	0.0635
d/mm	0.9774	0.0195	0.0174
high adaptative interval H	0.9 ~ 1	0 ~ 0.2	0 ~ 0.3

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表 3 响应面模型随机非样本点检验结果

Table 3. Test results of random non-sample points for the response surface model

number	σ/MPa	$ \hat \sigma $/MPa	stress error ε_σ/%	d/mm	$ \hat d $/mm	displacement error ε_d/%
1	112.7705	111.0160	1.56	576.5725	585.7213	1.59
2	112.7441	115.6223	2.55	515.2002	535.5826	3.96

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表 4 各算法的参数值

Table 4. Parameter values for each algorithm

algorithm	parameter	value
GA	type	real coded
	selection	roulette wheel (proportionate)
	crossover	whole arithmetic
		(probability = 0.8, α = [−0.5, 1.5])
	mutation	Gaussian (probability = 0.05)
PSO	cognitive and social constant inertia weight	(C₁, C₂): (2, 2)
PSO	cognitive and social constant inertia weight	linear reduction from 0.9 to 0.1
FA	α	0.5
	β	0.2
	γ	1
AOA	α	5
AOA	µ	0.5
IAOA	α	5
IAOA	µ	0.5

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表 5 不同算法抱杆优化结果对比

Table 5. Optimization results comparison by different algorithms for the holding pole

algorithm	b₁/mm	t₁/mm	b₂/mm	t₂/mm	b₃/mm	t₃/mm	W/kg	evaluation number
GA	146	11	40	10	40	10	40 721.26	1 000
PSO	150	12	40	10	40	5	40 773.77	1 000
FA	144	11	59	9	85	8	47 689.62	1 000
AOA	150	12	40	10	40	4	40 164.25	1 000
IAOA	150	10	40	10	40	10	39 780.55	1 000

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表 6 抱杆优化前后结果对比

Table 6. Results comparison before and after optimization for the holding pole

situation	b₁/mm	t₁/mm	b₂/mm	t₂/mm	b₃/mm	t₃/mm	W/kg	time T/s
initial	150	12	70	5	70	5	43 335.21	24.06
IAOA	150	10	40	10	40	10	39 780.55	6×10⁻⁵

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