求解目标失效概率对应的功能函数分位点是保障飞行器结构可靠性设计优化解耦算法效率的核心，而针对适航要求的极低目标失效概率叠加含高维随机输入功能函数的极端场景，目前还十分缺乏分位点的高效算法。为此，本文提出了该场景下分位点高效求解的周向-径向混合重要抽样法。所提方法首先利用目标失效概率与分位点的单调关系，将目标失效概率对应的分位点求解的逆问题转化为多个分位点对应的失效概率校准的正问题。然后构建参数化的周向 von-Mises-Fisher-径向 Nakagami混合密度(vMFNM)模型，以渐进分层的方式自适应逼近失效概率校准的最优重要抽样密度，同时，通过设计vMFNM模型的更新与信息共享策略，引导极端场景下分位点的高效搜索。算例结果表明，由于vMFNM模型对正向校准失效概率时方差的大幅缩减，以及共享vMFNM模型信息对分位点搜索的高效引导，从而使得所提方法明显优于已有的对比方法。
Evaluating the quantiles corresponding to target failure probabilities is central to efficiently decouple the reliability based design optimization of aircraft structure. However, efficient algorithms for quantile estimation are severely lacking for the extreme scenarios involving very low target failure probabilities random inputs required by airworthiness standards and high-dimensional. To address this, this paper proposes a circumferential-radial importance sampling based quantile evaluation (CRIS-QE) method. Leveraging the monotonic relationship between target failure probabilities and quantiles, the proposed CRIS-QE transforms the inverse problem of quantile evaluation into a forward problem of failure probability calibration at multiple thresholds. A parametric circumferential von-Mises-Fisher-radial Nakagami mixture model (vMFNM) is constructed to adaptively approach the optimal importance sampling density for failure probability calibration in a progressive and stratified manner. Concurrently, updating and information-sharing strategies for the vMFNM model are designed to guide the efficient search for quantiles in extreme scenarios. Numerical and engineering examples demonstrate that the proposed CRIS-QE significantly outperforms existing methods due to the substantial variance reduction provided by the vMFNM model and the efficient guidance offered by shared model information.
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