﻿ 基于改进的动态Kriging模型的结构可靠度算法<sup>*</sup>
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1. 北京航空航天大学 可靠性与系统工程学院, 北京 100083;
2. 北京航空航天大学 可靠性与环境工程技术重点实验室, 北京 100083

Structural reliability algorithm based on improved dynamic Kriging model
WEI Juan1,2, ZHANG Jianguo1,2, QIU Tao1,2
1. School of Reliability and Systems Engineering, Beihang University, Beijing 100083, China;
2. Science and Technology on Reliability and Environment Engineering Laboratory, Beihang University, Beijing 100083, China
Received: 2018-05-28; Accepted: 2018-08-24; Published online: 2018-09-07 13:54
Foundation item: National Natural Science Foundation of China (51675026)
Corresponding author. ZHANG Jianguo, E-mail: zjg@buaa.edu.cn
Abstract: For complex aerospace machinery products, the limit state functions are often implicit and highly nonlinear, and the reliability calculation usually requires time-consuming finite element analysis. In this paper, the particle swarm optimization-simulated annealing (PSOSA) algorithm is applied to the optimization of the correlation parameters of the dynamic Kriging model, which improves the prediction accuracy. At the same time, with the dynamic update mechanism, sample points are gradually added to reduce the number of function callsas much as possible, thereby improving the calculation efficiency. The algorithm is applied to the structural reliability analysis. The Monte Carlo method, response surface and other classic algorithms are compared, and the results of the proposed algorithm are closer to those of Monte Carlo method, and the calculation time is greatly shortened, which shows that the efficiency and accuracy of the algorithm are improved significantly.
Keywords: limit state function     dynamic update     Kriging model     particle swarm optimization-simulated annealing (PSOSA) algorithm     reliability

1 动态Kriging模型 1.1 Kriging模型

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1.2 动态更新机制

2 改进的动态Kriging模型 2.1 优化算法

2.1.1 PSO算法

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PSO的工作原理如下：首先，粒子群由搜索区域内随机分布的粒子组成，然后，PSO通过更新粒子的速度和位置来迭代寻找最优解。当满足收敛标准时，循环终止。但是，运用公式PbPg来搜索最优解，收敛速度快。然而所有粒子有移向最优经验点的趋势，并在局部区域内搜索，所以可能会导致收敛过早的问题。因此，尽管PSO比其他方法收敛速度快，但结果可能精度不高。

2.1.2 SA算法

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φj为每个变量接受的移动次数与试验次数的比值，则扰动可写为

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2.1.3 混合PSOSA算法

1) 惯性项(wvki)，使得粒子保持原来的速度向量，以防粒子的速度急剧变化。

2) 认知行为(c1R1(PkbXki))，包含粒子经历过的最优位置。

3) 社会行为(c2R2(PgXki))，给予粒子朝群体最优位置运动的趋势。

 图 1 PSOSA算法示意图 Fig. 1 Sketch map of PSOSA algorithm

1) 通过改进速度矢量的组成促使群体向整体最优位置所在的搜索空间移动，这加快了收敛过程。

2) PSO算法和SA算法的结合提高了整个寻优机制的质量和稳定性。

3) 在SA迭代中PSO速度公式中的认知成分得到了保留。

2.2 Kriging建模流程

1) 似然函数作为适应度函数。

2) 为转化为无约束优化，粒子采用对数形式。

 图 2 改进的动态Kriging模型流程图 Fig. 2 Flowchart of improved dynamic Kriging model
3 案例分析 3.1 数值算例1

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 方法 样本点 βr 失效概率/10-3 MCS 108 6.3 RSM 65 2.392 7 8.3 经典Kriging 40 2.475 1 6.7 PSO-Kriging 40 2.480 5 6.56 本文 40 2.489 3 6.4

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 图 3 极限状态函数 Fig. 3 Limit state function
 图 4 算法说明图 Fig. 4 Illustration of algorithm
3.2 数值算例2

 方法 函数调用次数 βr 失效概率/10-3 MCS 106 4.16 RSM 5 1.472 9 70.39 经典Kriging 30 2.624 5 4.30 PSO-Kriging 30 2.630 7 4.26 本文 30 2.639 6 4.15

3.3 工程案例

 参数 涡轮盘的转速ω 弹性模量(轮盘)E1 泊松比(轮盘)ε1 密度(轮盘)ρ1 弹性模量(销轴)E2 泊松比(销轴)ε2 密度(销轴)ρ2 均布载荷P 均值 9 550 r/min 123 GPa 0.33 4.48 g/cm3 219 GPa 0.3 7.76 g/cm3 24 925 N 变异系数 0.1 0.015 0.01 0.02 0.015 0.01 0.002 0.1

 图 5 应力和应变计算结果 Fig. 5 Calculation results of stress and strain

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 图 6 θ*的迭代过程 Fig. 6 Iteration process of θ*

 方法 函数调用次数 样本点 失效概率/10-3 MCS方法 105 3.300 本文算法 15 40 3.352

4 结论

1) 将PSOSA算法引入Kriging的构造过程，PSO算法的全局搜索能力及SA算法跳出局部极小值的能力相结合，克服了经典Kriging过程的局限性和对初始样本点的依赖，保证了极大似然意义下的最优相关参数的求解精度，使寻优过程更加高效和精确，从而有效保证了Kriging模型的预测精度。

2) 本文同时通过动态更新机制，尽可能地减少样本点和调用有限元等数值计算的次数，能够较好地解决功能函数隐式和高度非线性的问题，通过数值案例分析，相比于其他可靠度算法，能够有效提高可靠度计算的效率和精度。通过工程案例分析，本文所提算法可应用于工程实际问题，尤其是功能函数隐式且复杂的问题。有一定的工程实用价值。

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#### 文章信息

WEI Juan, ZHANG Jianguo, QIU Tao

Structural reliability algorithm based on improved dynamic Kriging model

Journal of Beijing University of Aeronautics and Astronsutics, 2019, 45(2): 373-380
http://dx.doi.org/10.13700/j.bh.1001-5965.2018.0301