Reduced order model-based uncertainty modeling of structures with localized response

This paper focuses on the introduction of uncertainty in reduced order models (ROMs) of a class of structures exhibiting a localized response. Considered first are the structures for which both the mean model, on which the ROM is constructed, and the uncertain ones exhibit a static response localized to the neighborhood of the excitation. A straightforward application to these ROMs of the maximum entropy framework is found to lead to a “globalization” of the response, consistently with the entropy maximization. Thus, an extension of the nonparametric stochastic modeling approach is developed here that is based on first splitting the ROM stiffness matrix into a component that induces the localized behavior and another propagating the response to the rest of the structure. The randomization of these two components is then carried out separately to maintain their respective, local and global, characters. This stochastic model involves three hyperparameters, one controlling the globalization of the uncertainty (δ_G), another governing the level of uncertainty in the localized zone (δ₁), and a third one controlling the distortion of the response in the localized zone (δ_L). This modeling isapplied to two example structures with different features for which the localization of the uncertain static response is demonstrated. It also shown that δ₁ is the key hyperparameter for such applications. The second part of this investigation focuses on structures, e.g., bladed disks, such that the mean model exhibits global mode shapes while those of the uncertain structures are strongly localized. A variation of the approach developed in the first part of the paper is developed that is shown to produce the expected localization of the mode shapes. In such applications, it is key to induce distortion of the free response and thus the hyperparameter δ_L is the dominant one.