Second-order time-variant reliability analysis method based on Hessian matrix approximation

基于Hessian矩阵逼近的二阶时变可靠性分析方法

¹ Beijing Institute of Astronautical System Engineering, Beijing 100076, China
² School of Astronautics, Northwestern Polytechnical University, Xi'an 710072, China

Received: 12 May 2024

Abstract

First-order time-variant reliability analysis(TRA) methods have been widely studied in the literature. However, these methods have low accuracy when solving time-variant reliability analysis problems with high nonlinear performance functions. To solve this problem, this paper proposes a second-order TRA method based on Hessian matrix approximation. First, the symmetric rank-one based quasi-Newton method is introduced to calculate the most probable point(MPP) and obtain approximated Hessian matrix of instantaneous performance function. Then, the instantaneous performance function is approximated by a second-order parabolic model, and therefore the instantaneous response is transformed into the weight sum of a normal variable and a chi-square variable. Correspondingly, the time-variant response is transformed into a non-stationary non-Gaussian stochastic process(NNP), which improves the accuracy of TRA. The Kriging models of the MPP trajectory and the mean curvature radius function are constructed, and the second-order time-variant reliability is estimated by sampling in the original random space to avoid the difficulty of modeling and simulation of the NPP in the response space. Finally, the results of two simulation examples show that the proposed method can significantly improve the accuracy of TRA without increasing the computational cost compared with first-order TRA method.

摘要

一阶时变可靠性分析方法在面对具有较高非线性时变功能函数的问题时, 分析精度较低。对此, 提出一种基于Hessian矩阵逼近的高精度二阶时变可靠性分析方法。将最大可能点搜索算法与基于对称秩一校正的Hessian矩阵迭代逼近方法结合, 避免直接求解Hessian矩阵; 建立瞬时功能函数的二阶抛物面模型, 将瞬时响应转化为正态变量与χ²分布随机变量的加权和形式, 将时变响应转化为非平稳非高斯随机过程(non-Gaussian non-stationary process, NNP), 提高分析精度; 建立最大可能轨迹和平均曲率半径函数的Kriging模型, 并通过在原始随机空间采样, 计算二阶时变可靠度, 避免在响应空间对NNP进行建模和数值模拟的困难。3个仿真算例结果表明, 所提方法可在不增加计算量的情况下显著提高时变可靠性分析精度。