A fully adaptive method for structural stochastic response analysis based on direct probability integral method

This paper proposes a fully adaptive method to estimate the probability density function (PDF) of the responses for stochastic structures under static and dynamic loads. The PDF is described by a recently published direct probability integral method (DPIM). In DPIM, the key smoothing parameter sigma and the number of samples Ns, which have significant impacts on the determination of PDF and the efficiency of stochastic response calculation, are still difficult to determine in a completely adaptive manner. To this end, we derive a weighted kernel density estimation (weighted-KDE) method to adaptively obtain the sigma by minimizing the mean integrated squared error between the estimated PDF and the real PDF. In addition, a new iterative sampling strategy is proposed to adaptively determine the Ns, in which a candidate sample pool is generated firstly, then the samples are gradually selected from the pool. This strategy ensures that the used samples in each iteration can fill the probability space in a 'highly uniform' manner. Based on the standard deviation calculated by the estimated PDF, a stopping criterion is presented to stop the iterative process. Moreover, the detailed steps of the proposed fully adaptive method is given by combining the weighted-KDE method and the sampling strategy. Four engineering examples validate the adaptive ability, efficiency and accuracy of the proposed method. (c) 2022 Elsevier B.V. All rights reserved.