A hyperspherical area integral method based on a quasi-Newton approximation for reliability analysis

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.