The First-Order Reliability Method (FORM) is renowned for its high computational efficiency, but its accuracy declines when addressing the nNar Limit-State Function (LSF). The Second-Order Reliability Method (SORM) offers greater accuracy; however, its approximation formula can sometimes introduce errors. Additionally, SORM requires extra calculations involving the Hessian matrix, which can reduce its efficiency. To balance efficiency and accuracy, a Hyperspherical Area Integral Method based on a Quasi-Newton Approximation (HAI-QNAM) for reliability analysis is proposed. This method initially employs a quasi-Newton method to determine the Most Probable Point (MPP) of failure, calculate the reliability index, and obtain the approximate Hessian matrix. Then, based on the Curved Surface Integral (CSI) method, the area of the approximate failure domain and the area of the hypersphere are obtained. Using the proportionality of their areas, the failure probability is then calculated. Finally, the proposed method's accuracy and efficiency are validated through examples.

中文翻译：

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一种基于准牛顿近似的超球面面积积分方法，用于可靠性分析


一阶可靠性方法 （FORM） 以其高计算效率而闻名，但在处理 nNar 极限状态函数 （LSF） 时，其准确性会下降。二阶可靠性方法 （SORM） 提供更高的准确性;但是，其近似公式有时会引入错误。此外，SORM 需要涉及 Hessian 矩阵的额外计算，这可能会降低其效率。为了平衡效率和精度，该文提出一种基于准牛顿近似（HAI-QNAM）的超球面面积积分方法进行可靠性分析。该方法最初采用准牛顿方法来确定失效的最大可能点 （MPP），计算可靠性指数，并获得近似的 Hessian 矩阵。然后，基于曲面积分 （CSI） 方法，得到近似失效域的面积和超球体的面积;然后利用它们面积的比例计算失效概率。最后，通过算例验证了所提方法的准确性和效率。