POD与MSM原理和应用的对比分析

为揭示本征正交分解(proper orthogonal decomposition, POD) 与模态叠加法(mode superposi-tion method, MSM) 在基本原理和工程应用中的异同,对POD 和MSM 的数理基础进行了对比性分析;通过风洞试验及动力响应时程分析获得了一座冷却塔的表面脉动风载荷及其脉动响应,以此两个时空随机场为例对POD 和MSM 两种方法的应用进行了对比性阐述. POD 和MSM 均通过引入空间模态和相应的时间坐标,经线性叠加来实现对时空随机场的分解与重建,但两者的应用对象并不一致:前者多用于已知随机载荷场的分析而后者多用于未知结构响应的计算. 两者空间模态和时间坐标的提取方法和数理意义以及各模态对原随机场的贡献等亦不相同. 尽管POD 和MSM 两种方法所得空间模态均有正交性,但时间坐标的正交性仅存在于POD 方法,故POD 各阶模态的本征值之和可以完全反应对原随机场的贡献,而MSM 的模态贡献则存在一定的耦合性.

COMPARATIVE STUDY OF THE PRINCIPLES AND APPLICATIONS OF POD AND MSM

The similarities and differences between the proper orthogonal decomposition (POD) and the mode superposition method (MSM) are studied not only in principles but also in applications. According to the principles of POD and MSM, comparative studies are made theoretically. The similarities and differences are illustrated in their respective applications, with two random fields of the fluctuating wind loads and the corresponding fluctuating wind-induced responses of a hyperboloidal cooling tower as examples, obtained from wind tunnel tests and the dynamic calculations in the time-domain. Both POD and MSM are methods used for the decomposition and reconstruction of a random field by linear superpositions of the spatial modes and corresponding time coordinates. However, their application fields are quite different: the POD is always used for the analysis of known load fields but the MSM is for the calculation of unknown response fields. Furthermore, the extraction methods of the spatial modes and the corresponding time coordinates for the POD and the MSM are different, as well as their mathematical and physical meanings and their contributions to the original random fields. The spatial modes of the POD and the MSM are both orthotropic, but while the time coordinates of the POD are orthotropic, those of the MSM are not. Consequently, the sum of the POD eigenvalues reflects the whole energy of the original random field, but it is not the case for the MSM due to the coupling between different modes.