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SummaryMultiquery problems such as uncertainty quantification (UQ), optimization of a dynamical system require solving a differential equation at multiple parameter values. Therefore, for large systems, the computational cost becomes prohibitive. This issue can be addressed by using a cheaper reduced order model (ROM) instead. However, the ROM entails error in the solution due to approximation in a lower dimensional subspace. Moreover, the ROM lacks robustness over a wide range of parameter values. To address these issues, first, an upper bound on the norm of the state transition matrix is derived. This bound, along with the residual in the governing equation, are then used to develop an error estimator for general nonlinear dynamical systems. Furthermore, this error estimator is used in conjunction with the modified greedy search algorithm proposed by Hossain and Ghosh (Int J Numer Methods Eng, 2018;116(12‐13): 741‐758) to adaptively construct a robust proper orthogonal decomposition‐based ROM. This adaptive ROM is subsequently deployed for UQ by invoking it in a statistical simulation. Two numerical studies: (i) viscous Burgers' equation and (ii) beam on nonlinear Winkler foundation, showed an improved accuracy of the error estimator compared to the current literature. A significant computational speed‐up in UQ is achieved.

Reduced order models (ROMs) are becoming increasingly useful for saving computational cost in response prediction of vibrating systems. In a number of applications such as uncertainty quantification, ROMs require robustness over a wide variation of parameters. Accordingly, often they are classified as local and global, based on their performance in the parametric domain. Availability of an error bound of a ROM helps in achieving this robustness, mainly by allowing adaptivity. In this work, for a linear random dynamical system, first, an a posteriori error bound is developed based on the residual in the governing differential equation. Next, based on this error bound, two adaptive methods are proposed for building robust ROMs, that is, one for local, and another for global. While both methods are based on a greedy search approach, a modification is proposed in the training stage of the global ROM for accelerated convergence. These methods are then applied to an uncertainty quantification problem in a statistical simulation framework, and accordingly, two algorithms are developed. A detailed numerical study on vibration of a bladed disk assembly is conducted to study the accuracy and efficiency of the proposed error bound and adaptive ROMs. It is found that these adaptive ROMs provide a considerable speed-up in estimating the probability of failure. KEYWORDS greedy search, probabilistic mechanics, proper orthogonal decomposition, reduced order model, stochastic structural dynamics Abbreviations: HDM, higher dimensional model; POD, proper orthogonal decomposition; ROM, reduced order model; UQ, uncertainty quantification. Int J Numer Methods Eng. 2018;116:741-758.wileyonlinelibrary.com/journal/nme

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