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Analytical approximation for stationary reliability of certain and uncertain linear dynamic systems with higher-dimensional output

机译：具有高维输出的某些不确定线性动力系统平稳可靠性的解析近似

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摘要

An analytical approximation for the calculation of the stationary reliability of linear dynamic systems with higher-dimensional output under Gaussian excitation is presented. For systems with certain parameters theoretical and computational issues are discussed for two topics: (1) the correlation of failure events at different parts of the failure boundary and (2) the approximation of the conditional out-crossing rate across the failure boundary by the unconditional one. The correlation in the first topic is approximated by a multivariate integral, which is evaluated numerically by an efficient algorithm. For the second topic some existing semi-empirical approximations are discussed and a new one is introduced. The extension to systems with uncertain parameters requires the calculation of a multi-dimensional reliability integral over the space of the uncertain parameters. An existing asymptotic approximation is used for this task and an efficient scheme for numerical calculation of the first-and second-order derivatives of the integrand is presented. Stochastic simulation using an importance sampling approach is also considered as an alternative method, especially for cases where the dimension of the uncertain parameters is moderately large. Comparisons between the proposed approximations and Monte Carlo simulation for some examples related to earthquake excitation are made. It is suggested that the proposed analytical approximations are appropriate for problems that require a large number of consistent error estimates of the probability of failure, as occurs in reliability-based design optimization. Numerical problems regarding computational efficiency may arise when the dimension of both the output and the uncertain parameters is large.
机译：提出了在高斯激励下具有高维输出的线性动力系统平稳可靠性计算的解析近似。对于具有某些参数的系统，讨论了两个主题的理论和计算问题：（1）失效边界不同部分的失效事件的相关性；（2）无条件越过失效边界的条件越界率的近似值一。第一个主题中的相关性通过多元积分来近似，该多元积分由有效算法进行数值评估。对于第二个主题，讨论了一些现有的半经验近似值，并引入了一个新的近似值。对具有不确定参数的系统的扩展要求在不确定参数的空间上计算多维可靠性积分。现有的渐近逼近用于此任务，并提出了一种用于对被积数的一阶和二阶导数进行数值计算的有效方案。使用重要性抽样方法的随机模拟也被视为替代方法，特别是对于不确定参数的大小适中的情况。对于与地震激发有关的一些示例，对建议的近似值和蒙特卡洛模拟进行了比较。建议的建议是，近似分析适用于需要大量一致的故障概率误差估计的问题，如在基于可靠性的设计优化中出现的情况。当输出和不确定参数的尺寸都很大时，可能会出现与计算效率有关的数值问题。

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