非均布随机参数结构非线性响应的概率密度演化简

首先考察了概率密度演化理论中的点演化和群演化与概率空间剖分的关系. 继而，讨论了点集筛选的基本准则. 在此基础上推广了点集偏差的概念，对非均匀、非正态的一般多维分布，提出了广义F 偏差（GF 偏差）的概念，避免了偏差计算的NP 难解问题. 探索了GF 偏差与EF 偏差的关系. 以GF 偏差最小化为准则，建议了概率空间最优剖分与点集重整的新策略. 结果表明，上述方法能够处理包含多达数10 个随机变量的结构动力响应概率密度演化分析问题. 最后，指出了需要进一步研究的问题.

PROBABILITY DENSITY EVOLUTION ANALYSIS OF NONLINEAR RESPONSE OF STRUCTURES WITH NON-UNIFORM RANDOM PARAMETERS

The probability density evolution method (PDEM) provides a feasible approach for nonlinear stochastic response analysis of multi-degree-of-freedom systems. In the present paper, the point evolution, ensemble evolution and the partition of probability-assigned space are firstly revisited. The criterion for point selection is then explored. The concept of generalized F-discrepancy (GF-discrepancy), which avoids the NP-hard problem of computation, is introduced for random variables of general non-uniform, non-Gaussian distribution as an index to measure the quality of a point set. The relationship between GF-discrepancy and EF-discrepancy is explored and the error bound is studied by the extended Koksma-Hlawka inequality. Based on the GF-discrepancy, a new strategy for point-selecting and space-partitioning is proposed. The numerical example shows that the proposed method enables highly accurate probability density evolution analysis of nonlinear structures involving dozens of non-uniform random variables. Problems to be further studied are discussed.