失效概率函数求解的高效算法

结合基于分数矩约束的极大熵方法和替代模型法，发展了一种失效概率函数求解的高效算法。所提算法的基本思路是利用自主学习的迭代Kriging方法来构造失效概率函数，即采用较少的训练样本来构造粗糙的失效概率函数，在此基础上通过添加新的违反学习函数约束的样本来更新失效概率函数，直到达到精度要求。对于每一个分布参数的训练样本点，所提方法采用分数矩约束的极大熵法来求解相应的失效概率样本。由于分数矩的计算采用了高效的降维积分，并且由于分数矩约束下极大熵法中优化策略高效地逼近了响应的概率密度函数，从而使得失效概率样本能够被高效高精度地估计出来。为了检验所提方法的精度及效率，给出了两个算例，对比了所提方法与已有的失效概率函数求解的Bayes公式法及Monte Carlo法等，结果表明，所提方法适用于求解复杂的功能函数问题，且在满足精度要求的基础上大大降低了计算量。
An efficient method was developed to obtain the failure probability function which combines the fractional moment-based maximum entropy method and the surrogate model method. The idea of the process is to build the failure probability function iteratively by the active learning Kriging method. Firstly, a crude failure probability function was established by using a few training samples. Then the training samples which violate the restraints of the learning function were added to update the failure probability function until the accuracy of the problem was satisfied. The fractional moment-based maximum entropy method was used to get the failure probability sample for every distribution parameter′s training sample. The samples of the failure probability could be evaluated efficiently and accurately for the optimization strategy in the fractional moment based maximum entropy method, which could approximate the probability density function of the response effectively, and the fractional moments were estimated by the dimensional reduction method. Two examples were illustrated in the end to compare several methods such as the Bayes method, the Monte Carlo method, and so on. From the numerical results, it can be seen that the proposed method can accurately solve the problem with complex performance function and can reduce the computational cost significantly.