地震动随机场的POD降维表达

POD-based dimension reduction representation of stochastic ground motion fields

本文从本征正交分解(POD)的基本理论出发,推导了非平稳随机向量过程的两类POD表达,即基于正交随机变量的POD表达和基于随机相位角的POD表达.澄清了基于随机相位角的POD表达是基于正交随机变量的POD表达的一个特例,并从基本随机变量的数量和约束条件两个方面剖析了它们之间的差异,发现通过施加约束条件可以有效地降低随机向量过程模拟的随机度(基本随机变量的数量).为此,通过对基于正交随机变量的POD表达施加更强的约束条件,实现了仅用2个基本随机变量即可对非平稳随机向量过程进行高效降维模拟,进而可在全概率信息上精细地表达非平稳随机向量过程.通过对地震动随机场模拟的数值分析,得到了非平稳地震动加速度向量过程的均值、标准差、演变功率谱及相关函数等均与目标值拟合一致的良好结果,说明了本文方法的有效性.同时,为了进一步从地震动的工程特性方面验证本文方法的适用性,根据《中国地震动参数区划图》中划分的5种场地类别,选取920条实测强震动记录,从地震动加速度的反应谱和傅里叶幅值谱两个方面进行了对比分析.研究表明,本文方法模拟结果与实测强震动记录具有很好的一致性,这为本文方法的工程应用奠定了坚实基础.

In this paper, two kinds of proper orthogonal decomposition(POD) representations for simulating the non-stationary stochastic vector processes are derived based on the basic theory of POD, namely the orthogonal-random-variables-based POD representation and the random-phase-angles-based POD representation. It is clarified that the random-phase-angles-based scheme is a special case of the orthogonal-random-variables-based scheme, and the differences between them are analyzed from the number of elementary random variables and the constraint conditions. It is found that the randomness degree(the number of elementary random variables) of the non-stationary stochastic vector processes could be greatly reduced by introducing some appropriate constraint conditions. To this end, the efficient dimension reduction simulation of the non-stationary stochastic vector processes with just two elementary random variables is realized through introducing more rigorous constraints on the orthogonal random variables in the orthogonal-randomvariables-based scheme. Also, the complete probability information of the stochastic processes is fully emerged. Through the numerical analysis of the ground motion stochastic field simulation, it is obtained that the relative errors of the mean and standard deviation, evolutionary power spectrum and coherence function of the non-stationary ground motion acceleration vector process are all in good agreement with the corresponding target values, which shows that the proposed method is effective and accurate.Meanwhile, in order to further verify the applicability of this method in terms of engineering characteristics upon ground motion,according to the five classes of the typical soil sites divided in "Seismic ground motion parameters zonation map of China", 920 strong motion records are selected for comparing and analyzing the response spectrum and the Fourier amplitude spectrum of ground motion acceleration, respectively. The research shows that the simulation results of the proposed method have good consistency with the strong motion records, which lays a solid foundation for this proposed method applied in engineering applications.