表4 算例二各随机变量/参数信息[12]
Tab.4 Random variables/parameters of example 2 图4 弹簧及其受力示意图[]
18
Fig.4 Illustration of the spring and force diagram
其中, Q为弹簧闭合端圈数, Q= 2。已知在实际生产中,平均中径D和丝径d具有相关性,故在求解RBDO问题时需将其考虑进来。Dd、 的 300组样本见图5,对其进行相关性分析,结果见表5。从结果中可知, Clayton Copula 的AIC值最小,故D和d之间的最优 Copula 为 Clayton Copula ,其参数θ= 2.92。若用Gaussian Copula对样本进行拟合,则对应参数θ= 0.76。
图5 Dd、的300组样本
Fig.5 300 samples of Dd、
表5 D、d之间备选Copula参数估计值及其AIC值Tab.5 Estimate of Copula parameter and AIC value
between D、d 与算例一类似,虽然已经得到最优Copula 函数,本文仍在以下3种相关情形下进行RBDO求解: ①Clayton Copula, θ= 2.92;②Gaussian Copu⁃ la, θ= 0.76;③相互独立。得到的优化结果见表6。从表6中可以看出,变量之间的相关情形对优化的结果影响较大。对于同样的一组样本,用不同的Copula函数拟合将得到不同的优化结果。此外,在3种相关情形下,优化均经过3个迭代步达到收敛,进一步说明了本文方法具有较好的收敛性和计算效率,迭代过程见图6。
表 6 算例二优化结果
Tab.6 Optimization results of example 2 图6 算例2迭代历史
Fig.6 Iteration history of example 2
现假设 X2 和 X3之间实际的 Copula 函数为Clayton Copula,参数θ= 2.92,用 Monte ⁃ Carlo 模拟法对3种优化结果进行可靠性验证(即用Monte⁃ Carlo法验证时产生的样本之间的相关类型为Clayton Copula),其可靠度和失效概率见表7。由表7可知:若将X2和 X3的样本用Gaussian Copula拟合,得到的优化结果中g1的可靠度β仅为 2.330 9,失效概率Pf达到了0.009 9,是目标值( 0.001 3)的近8倍,未能满足可靠性;将X2和X3错误地视为表7 实际相关情形为Clayton Copula时,
各优化结果可靠度
Tab.7 Reliability of optimization results when the actual correlation is Clayton Copula
独立的随机变量,其优化结果的可靠度远大于目标可靠度,过于保守。前文提到, Gaussian Copula实际上等效为传统的Nataf变换,即只考虑了变量间的线性相关。从该算例中可以看出,仅用线性相关系数对随机变量的相关性进行描述是不足够的,若不考虑随机变量间存在的复杂非线性相关性,可能会得到不可靠的优化结果。将Copula 函数引入到RBDO中,通过AIC准则选择出正确的Copula函数能充分考虑到变量间的非线性相关性,提高优化结果的精度。
4 结论
本文基于 Copula函数提出了一种RBDO 算法,为求解存在复杂非线性相关性的结构可靠性优化设计问题提供了有效工具。首先根据数据样本通过极大似然法和AIC准则选择出最优Copu⁃ la;其次,由最优Copula函数和各变量的边缘分布构建出联合概率分布函数;最后,将联合概率分布函数用于可靠性分析,从而求解RBDO问题。两个数值算例验证了本文方法的有效性,结果表明: Copula函数的类型对优化结果有较大影响;在某些存在复杂相关性的情况下,使用传统的Nataf 变换可能得到不可靠的优化结果;通过Copula函数,能够描述变量间的非线性相关和尾部相关性,从而提高优化结果的精度。此外,本文方法主要针对二维相关情形,未来可引入Vine⁃Copula等模型对本文方法进行拓展,从而求解变量间存在复杂多维相关性的RBDO问题。
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