粒子群优化与Kriging模型相结合的结构非概率可靠性分析

乔心州; 陈永婧; 刘鹏; 方秀荣

doi:10.21656/1000-0887.420308

粒子群优化与Kriging模型相结合的结构非概率可靠性分析

doi: 10.21656/1000-0887.420308

西安科技大学机械工程学院，西安 710054

基金项目: 陕西省自然科学基础研究计划(2019JQ-796)

详细信息

作者简介:
乔心州（1974—），男，副教授，硕士生导师（通讯作者. E-mail：qiaoxinzhou@xust.edu.cn）

中图分类号: O213.2
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- 文章访问数: 519
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- 被引次数: 0
出版历程
- 收稿日期: 2021-10-11
- 录用日期: 2022-03-10
- 修回日期: 2021-12-08
- 网络出版日期: 2022-11-03
- 刊出日期: 2022-12-01

Non-Probabilistic Structural Reliability Analysis Integrating the PSO and the Kriging Model

College of Mechanical Engineering, Xi’an University of Science and Technology, Xi’an 710054, P.R.China

摘要

摘要:
针对复杂结构可靠性分析中面临的隐式功能函数和小样本问题，提出了一种粒子群优化和Kriging模型相结合的结构非概率可靠性分析方法。采用多维椭球描述结构不确定参数，运用粒子群优化对模型相关参数进行求解，并构建隐式功能函数的Kriging模型进行可靠性分析。三个算例结果表明所提方法有效可行，精度和效率均优于基于Kriging模型的非概率可靠性分析方法。
- 粒子群优化 /
- Kriging模型 /
- 非概率可靠性分析
Abstract:
In view of the problems with implicit performance functions and limited experimental data in the reliability analysis of complex structures, a non-probabilistic reliability method combining the particle swarm optimization (PSO) with the Kriging model was presented. A multidimensional ellipsoid was first used to characterize the uncertain parameters of structures. The Kriging model was then constructed for the implicit performance function, wherein its optimal related parameters was determined through the PSO. Based on the proposed model the reliability analysis was explicitly conducted. The results of 3 numerical examples show that, the proposed method is of effectiveness and feasibility, and has higher accuracy and efficiency than those based on the traditional Kriging model.
- particle swarm optimization /
- Kriging model /
- non-probabilistic reliability analysis

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图 1 线性变换与可靠性指标

Figure 1. The linear transform and the reliability index

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图 2 算法流程图

Figure 2. The flowchart for the proposed method

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图 3 相对误差对比图（算例1）

Figure 3. Comparison of relative errors (example 1)

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图 4 链轮

Figure 4. A chain wheel

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图 5 相对误差对比图（算例2）

Figure 5. Comparison of relative errors (example 2)

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图 6 25杆桁架

Figure 6. A 25-bar truss structure

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图 7 相对误差对比图（算例3）

Figure 7. Comparison of relative errors (example 3)

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表 1 相关系数为0.2的迭代过程

Table 1. The iterative process for a correlation coefficient of 0.2

iteration step	design point	reliability index
0	(−2.4442, 4.2332)	4.8882
1	(−1.8607, 4.1064)	4.5083
2	(−1.8641, 4.1039)	4.5074
reference value	(−1.8692, 4.1017)	4.5075

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表 2 不同相关系数的可靠性指标（算例1）

Table 2. Reliability indexes for different correlation coefficients (example 1)

correlation coefficient	PSO-Kriging	iterations	CPU time	Kriging	iterations	CPU time	reference value
0	4.1704	2	0.45703 s	4.1653	5	1.00112 s	4.1701
0.2	4.5073	2	0.38524 s	4.5002	6	1.20133 s	4.5075
0.5	5.2134	3	0.68554 s	5.2039	5	1.01367 s	5.2138
0.7	5.9237	3	0.88563 s	5.9355	7	1.40156 s	5.9235
0.9	7.0337	2	0.57889 s	7.0302	6	1.16778 s	7.0339

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表 3 相关系数为0.9的迭代过程

Table 3. The iterative process for a correlation coefficient of 0.9

iteration step	design point	reliability index
0	(0.0662, 0.0265, 0.0088, 0.00650.4059)	0.4123
1	(−0.0031, −0.0004, 0.0146, 0.0141, 0.3458)	0.3464
2	(−0.0040, −0.0005, 0.0124, 0.0176, 0.3415)	0.3422
3	(−0.0044, −0.0007, 0.00132, 0.0179, 0.3441)	0.3448
4	(−0.0042, −0.0007, 0.0134, 0.0181, 0.3439)	0.3447
reference value	(−0.0058, −0.0015, 0.0146, 0.0155, 0.3437)	0.3444

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表 4 不同相关系数的可靠性指标（算例2）

Table 4. Reliability indexes for different correlation coefficients (example 2)

correlation coefficient	PSO-Kriging	iterations	CPU time	Kriging	iterations	CPU time	reference value
0	0.3469	4	16.15572 s	0.3406	7	27.16953 s	0.3465
0.3	0.3504	5	20.19465 s	0.3466	9	34.93226 s	0.3509
0.5	0.3517	4	17.23898 s	0.3612	7	26.35562 s	0.3513
0.7	0.3487	5	21.56773 s	0.3454	8	31.05089 s	0.3490
0.9	0.3447	4	18.06577 s	0.3404	7	28.09947 s	0.3444

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表 5 MCS求解的95%置信区间（算例2）

Table 5. The 95% confidence intervals of MCS solutions (example 2)

MCS value	mean value	deviation	lower bound	upper bound
0.3465	0.3469	0.0029	0.3463	0.3475
0.3509	0.3506	0.0029	0.3500	0.3511
0.3513	0.3511	0.0030	0.3505	0.3517
0.3490	0.3491	0.0030	0.3485	0.3497
0.3444	0.3448	0.0031	0.3442	0.3454

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表 6 不确定变量的分布参数

Table 6. The distribution parameters of uncertain variables

uncertain variable	mean value	radius
$ {A_1} $/mm²	700	70
$ {A_2} $/mm²	6200	620
$ {A_3} $/mm²	5300	530
$ {A_4} $/mm²	8800	880
$ L $/mm	15000	1500

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表 7 相关系数为0的迭代过程

Table 7. The iterative process for a correlation coefficient of 0

iteration step	design point	reliability index
0	(−0.0083, −0.4266, −0.3646, −0.0519, 0.8256)	0.9997
1	(−0.0109, −0.5652, −0.4831, −0.0587, 0.9338)	1.1952
2	(−0.0100, −0.5201, −0.4446, −0.0540, 0.8593)	1.0997
3	(−0.0089, −0.4827, −0.4126, −0.0478, 0.7875)	1.0870
4	(−0.0084, −0.5107, −0.4222, −0.0509, 0.8504)	1.0793
5	(−0.0105, −0.5065, −0.4289, −0.0465, 0.8372)	1.0694
6	(−0.0094, 0.5032, −0.4301, −0.0431, 0.8356)	1.0669
reference value	(−0.0092, −0.5043, −0.4280, −0.0484, 0.8340)	1.0656

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表 8 不同相关系数的可靠性指标（算例3）

Table 8. Reliability indexes for different correlation coefficients (example 3)

correlation coefficient	PSO-Kriging	iterations	CPU time	Kriging	iterations	CPU time	reference value
0	1.0669	6	27.02193 s	0.9944	9	37.73403 s	1.0656
0.3	1.2683	7	31.52558 s	1.1793	10	41.92578 s	1.2679
0.5	1.4906	6	28.02193 s	1.3997	10	40.44356 s	1.4917
0.7	1.9043	6	26.02193 s	1.8253	9	38.37882 s	1.9023
0.9	3.1439	7	31.52558 s	2.9934	11	46.11937 s	3.1431

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表 9 MCS求解的95%置信区间（算例3）

Table 9. The 95% confidence intervals of MCS solutions (example 3)

MCS value	mean value	deviation	lower bound	upper bound
1.0656	1.0659	0.0029	1.0653	1.0665
1.2679	1.2672	0.0028	1.2667	1.2679
1.4917	1.4919	0.0025	1.4914	1.4924
1.9023	1.9023	0.0028	1.9017	1.9029
3.1431	3.1430	0.0029	3.1424	3.1435

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