A New Framework for Model Reduction of Complex Nonlinear Dynamical Systems

Abstract

Considerable progress in computing technology in the past decades did not alleviate difficulty inherent in simulating complex dynamical systems. Reduced order models (ROMs) can be used to unburden these systems of redundant computations. While a variety of methods have been developed for reduced order modeling, they cannot be used for parametric study of nonlinear and complex systems, wherein we constantly change the parameters, input values, and energy levels. Parametric study is essential to determine the dynamics of complex systems. Only robust and persistent reduced order models, which remain stable with these changes, can be used for parametric study. In this dissertation, we develop a framework which measures the robustness and persistency of reduced order models. The framework quantifies the changes in the reduced models and singles out the most robust and persistent ones. The main advantage of this methodology is that it is applicable to the majority of databased model reduction methods. The approach begins with specifying a range of system's initial states, parameters, and inputs for the parametric study. The data is collected from simulations of the system with the parameters chosen randomly within that range. The dominant structures of data are then identified using the multivariate analysis methods such as proper orthogonal decomposition (POD) and smooth orthogonal decomposition (SOD). The framework identifies the persistent and robust structures and combines them to obtain the models suitable for parametric study within the specified range. Our aim is to investigate the fidelity of the framework for persistent model order reduction of large and complex dynamical systems. The framework is validated using several numerical examples including a large linear system and two complex nonlinear systems with material and geometrical nonlinearities. While the method is used for identifying the robust subspaces obtained from both POD and SOD, the results show that SOD outperforms POD in terms of stability, accuracy, speed, and robustness. Also, showing that SOD-based ROMs are robust, we no longer need to simulate full-scale models for many parameters. We only need to do few simulations using the full-scale model to build ROMs. In addition, we extend the application of the proposed approach to model order reduction of nonlinear control systems. We use SOD to identify the dynamically relevant modal structures of the control system. The identified SOD subspaces are used to develop persistent ROMs. Performance of the resultant SOD-based ROM is compared with POD-based ROM by evaluating their robustness to the