Double-loop surrogate model for time-dependent reliability analysis based on NARX and Kriging models
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摘要:
针对具有动态输出性能的结构系统，传统的求解时变可靠性的代理模型方法在建模时只关注当前瞬间作用于系统的随机变量的作用，而忽视了时间累积效果，使得模型对于时变可靠性的预测效果并不理想。基于此，提出了一种基于带外生输入的非线性自回归(NARX)模型和Kriging模型的时变可靠性分析的双层代理模型方法。所提方法在内层利用NARX模型构建给定随机输入变量下输出响应随时间的变化模型，准确模拟系统的动态行为；在外层基于NARX所得极值构建系统极值与随机变量之间的Kriging模型，得到时变结构系统的可靠性。通过3个算例验证了所提方法在处理具有较强波动性输出系统的可靠性问题时的有效性和准确性。
Abstract:Structural systems with dynamic output performance has gained more and more attention in engineering practice. However, most existing surrogate models for estimating time-dependent reliability of such systems only consider the effect of the random variables which are acting on the system at the current moment, but ignores their effect with time-dependent accumulation. Therefore, these models cannot give an accurate prediction for time-dependent reliability of the dynamic systems. To solve this problem. this paper proposes a double-loop surrogate model method for time-dependent reliability analysis based on the nonlinear autoregressive with exogenous input (NARX) model and Kriging model. In the inner loop of the proposed method, NARX model is used to describe the variation of the output response with time under given random input variables, which can accurately simulate the dynamic behavior of the system. In the outer loop, the Kriging model of the extreme value of the dynamic systems is built based on the random input samples and the corresponding extreme values predicted by the inner NARX model. The reliability of the time-varying structure system then can be easily obtained based on the outer Kriging model. Finally, the effectiveness and accuracy of the proposed method for reliability analysis of dynamic structural systems with fluctuating outputs is verified by three examples.
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Key words:
- time-dependent reliability /
- failure probability /
- surrogate model /
- NARX model /
- Kriging model
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图 2 NARX模型与Kriging模型预测的时变均值和标准差与参考解的对比
Figure 2. Comparison of reference resolution with time-dependent mean and standard deviation predicted by NARX model andKriging model
图 3 NARX模型与Kriging模型预测的时变均值和标准差与参考解的对比
Figure 3. Comparison of reference solution with time-dependent mean and standard deviation predicted by NARX model a nd Kriging model
图 6 NARX模型与Kriging模型预测的时变均值和标准差与参考解的对比
Figure 6. Comparison of reference solution with time-dependent mean and standard deviation predicted by NARX model and Kriging model
表 1 随机变量分布参数
Table 1. Variables and parameters
分布
类型运算
输出$\sigma_{{\rm{s}}}/({\rm{N} } \cdot {\rm{m} }^{-2})$ $A /{\rm{m}}^2$ ${J/{\rm{m} }^{4} }$ $n_0/{\rm{s}}^{-1}$ $\rho/({\rm{kg} }\cdot {\rm{m} }^{-3})$ $C$ 正态
分布均值 $1.1 \times {10^9}$ $0.012\;4$ $ 1.22 \times {10^{ - 4}} $ $163$ $8\;240$ $5.67$ 标准差 $1.1 \times {10^8}$ $1.24 \times {10^{ - 3}}$ $1.22 \times {10^{ - 5}}$ $16.3$ $824$ $0.567$ 
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表 2 不同方法的时变可靠性分析结果
Table 2. Time-dependent reliability analysis results of different methods
方法 $ P_{f} $ $\operatorname{Cov}(P_{f})$ 样本量 构建内层模型平均时刻点 构建外层模型样本量 误差/% NARX-Kriging $8.00 \times {10^{ - 5}}$ $0.035\;4$ $172$ $6$ $27$ $2.56$ Double-Kriging $8.76 \times {10^{ - 5}}$ $0.033\;8$ $196$ $5$ $39$ $6.70$ MC $8.21 \times {10^{ - 5}}$ $0.034\;9$ $7 \times {10^9}$
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表 3 分布参数
Table 3. Variables and parameters
分布类型 运算输出 $\varsigma$ $\omega_{\rm{n}}/({\rm{rad} }\cdot {\rm{s} }^{-1})$ $\alpha$ $A/{\rm{N}}$ $\varphi/({\rm{rad}}\cdot {\rm{s}}^{-1})$ 正态分布 均值 $0.02$ $2\text{π}$ $50$ $1$ $\text{π}$ 标准差 $0.001$ $0.1\text{π}$ $2.5$ $0.05$ $0.05\text{π}$
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表 4 各种方法的时变可靠性分析结果
Table 4. Results of each method
方法 $P_{f}$ $\operatorname{cov}(P_{f})$ 样本量 构建内层模型平均时刻点 构建外层模型样本量 误差/% NARX-Kriging $7.15 \times {10^{ - 4}}$ $0.083\;5$ $9\;900$ $150$ $67$ $1.93$ Double-Kriging MC $7.35 \times {10^{ - 4}}$ $0.082\;4$ $6 \times {10^8}$
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表 5 分布参数
Table 5. Variables and parameters
运算输出 $l_{1}/{\rm{m}}$ $h_{1}/{\rm{m}}$ $h_2 /{\rm{m}}$ $ A_{l} $ $\psi (t)/({\rm{m} }\cdot {\rm{s} }^{-1})$ 均值 $0.22$ $0.025$ $0.019$ $0.030$ $ {\mu _\psi }\left( t \right) $ 标准差 $0.002\;2$ $0.000\;25$ $0.000\;19$ $0.000\;30$ $ {\sigma _\psi }\left( t \right) $ 自相关函数 $ {\rho _\psi }\left( {{t_1},{t_2}} \right) $ 注：l1，h1，h2，Al为正态分布；${\psi}\left( t \right)$为随机过程。
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表 6 各种方法的时变可靠性分析结果
Table 6. Results of each method
方法 Pf Cov(pf) 样本量 构建内层模型平均时刻点 构建外层模型样本量 误差/% 计算结果 NARX-Kriging 6.32×10-4 0.056 3 7 640 40 191 2.27 Double-Kriging MC 6.18×10-4 0.056 8 6×108
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