###### Abstract  
Complex dynamic behavior of nonlinear structures makes it challenging for uncertainty analysis through Monte Carlo simulations (MCS). Surrogate modeling presents an efficient and accurate computational alternative for a large number of MCS. The previous study has demonstrated that the multi-input multi-output nonlinear autoregressive with exogenous input (MIMO-NARX) model provides good discrete-time representations of deterministic nonlinear multi-degree-of-freedom (MDOF) structural dynamic systems. Model order reduction (MOR) is executed to eliminate insignificant modes to reduce the computational burden due to too many degrees of freedom. In this study, the MIMO-NARX strategy is integrated with different meta-modeling techniques for uncertainty analysis. Different meta-models including Kriging, polynomial chaos expansion (PCE), and arbitrary polynomial chaos (APC) are used to surrogate the NARX coefficients for system uncertainties. A nine-DOF structure is used as an MDOF dynamic system to evaluate different meta-models for the MIMO-NARX. Good fitness of statistical responses is observed between the MCS results of the original system and all surrogated MIMO-NARX predictions. It is demonstrated that the APC-NARX model with the advantage of being data-driven is the most efficient and accurate tool for uncertainty quantification of nonlinear structural dynamics. nonlinear autoregressive with exogenous input (NARX); multi-degree-of-freedom; nonlinear structure; uncertainty analysis; meta-model +
Footnote †: journal: Medical  
significantly increases for accurate dynamic time-history simulations. The complex behavior exhibited by nonlinear systems brings great challenges to uncertainty analysis due to the requirement of a large number of samples for MCS. Dynamic analysis of a deterministic nonlinear structure can be considered as solving differential equations in the discrete-time domain. Nonlinear autoregressive with exogenous input (NARX) models provide good discrete-time representations of system dynamics as a function of previous response values as well as current and/or previous input signals [10; 11]. The NARX model thus has been successfully applied for describing nonlinear dynamic behavior for single-degree-of-freedom (SDOF) systems [12]. In the case of multi-degree-of-freedom (MDOF) systems, the single-input single-output (SISO) strategy establishes a separate NARX model for each output of the MDOF system. Multi-input multi-output (MIMO) formulation considers the inevitable response coupling between the multiple outputs of the NARX model [13; 14]. Nevertheless, a large number of degrees of freedom imposes significant computational demands for building the NARX model through the MIMO strategy. The model order reduction (MOR) method is usually utilized to reduce the degrees of freedom for the MIMO-NARX model, which has been demonstrated to significantly improve computational efficiency with little effect on its accuracy [15; 16; 17]. The NARX model can capture the nonlinear dynamic behavior of the deterministic system. A stochastic NARX model is characterized by a set of specified NARX model terms and associated random coefficients [18]. Once the NARX model is built from a series of nonlinear dynamic responses, the NARX coefficients need to be calibrated based on the specific system conditions including structural property and external excitation. This means that the NARX coefficients vary for different nonlinear dynamic behavior due to system uncertainties such as ground motions and structural properties. In uncertainty analysis, the nonlinear dynamic behavior of an MDOF system from a large number of MCS is difficult to obtain due to the unbearable computational cost of dynamic analysis. It means that the NARX coefficients cannot be calibrated directly by the target dynamic behaviors. Meta models have been used to govern the uncertainties in the nonlinear system represented by SISO-NARX in previous research [18; 19]. Different from the [PERSON] et al.