结构随机反应概率密度演化分析的数论选点法

密度演化方法可以直接获取结构的线性和非线性响应概率密度函数解答及其演化过程。当结构参数与激励中含有多个随机变量时，在多维随机变量空间中的离散代表点选点规则对密度演化分析的精度和效率至关重要。基于高维数值积分的数论方法，建议了多维随机变量空间的数论选点方法。利用多维随机变量空间的联合概率密度函数的球对称性或近似辐射衰减性质，对数论方法给出的单位超立方体中的分布点集进行筛选，可大幅度减少选点数目，从而将具有多个随机变量的结构随机响应分析问题计算工作量降低到与单一随机变量结构随机响应分析问题相当的水平。

Strategy of selecting points via number theoretical method in probability density evolution analysis of stochastic response of structures

The newly developed probability density evolution method (PDEM) is capable of capturing instantaneous probability density function and its evolution of linear and/or nonlinear stochastic response of structures. In the occasions that multiple random parameters are involved in the structural properties and external excitations, the strategy of selecting representative points required in the PDEM is of paramount importance to the accuracy and efficiency. Enlightened by the Number Theoretical Method successfully employed in high-dimensional numerical integration, the strategy of selecting points via Number Theoretical Method is proposed in the present paper. Further, making use of the spherically symmetric properties or the radial attenuation properties of the joint probability density function, the points scattered over the multi-dimensional hypercube selected by the Number Theoretic Method are sieved once again such that only the points inside the multi-dimensional hyper-ball are retained. With the proposed strategy of selecting points, the stochastic response analysis involving multiple random parameters is almost as efficient as the problem involving only one single random parameter.