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Date of Award


Document Type

Campus Access Dissertation

Degree Name

Doctor of Philosophy in Mechanical Engineering-Engineering Mechanics (PhD)

Administrative Home Department

Department of Mechanical Engineering-Engineering Mechanics

Advisor 1

Hassan Masoud

Committee Member 1

Gordon Parker

Committee Member 2

Susanta Ghosh

Committee Member 3

Yousef Darestani


Understanding and characterizing complex dynamical behaviors in nonlinear systems require the identification of their intrinsic coordinates or modes, a challenge distinct from linear systems where modal transformation is universal. Koopman operators and nonlinear normal modes (NNMs) serve as essential tools in the study of nonlinear dynamics. Koopman operators offer a method for depicting nonlinear dynamics within a linear framework. On the other hand, NNMs, which are an extension of linear normal modes (LNMs), are designed to capture the inherent invariance properties present in nonlinear dynamics. Because of the challenges involved in representing the Koopman operator and NNMs analytically, utilizing data-driven approaches could be a suitable solution for approximating the nonlinear modal spaces of nonlinear dynamical systems. In this dissertation, our goal is to utilize data-driven methods to approximate these two nonlinear modal analysis techniques for different nonlinear dynamical systems, such as fluid flows. Additionally, we will explore data-driven approaches for representing parametric nonlinear dynamical systems as data-driven surrogate models are suitable alternatives for modeling of parameterized nonlinear dynamical systems.

The second chapter of this dissertation explores the representation capabilities of Koopman operators and nonlinear normal modes (NNMs) for nonlinear dynamical systems. Addressing the realistic absence of closed-form models, we introduce a physics-integrated deep learning-based approach to identify Koopman eigenfunctions and NNMs. We assess their representation accuracy by reconstructing and predicting nonlinear dynamical responses. Numerical experiments on Duffing systems reveal that NNMs outperform Koopman operators in accuracy and computational efficiency.

The observation of significant coherent structures in fluid flows emphasizes the importance of discovering appropriate coordinates or modes. This discovery is essential for understanding and characterizing the complex dynamic features of fluid flows, which are inherently highly nonlinear systems. In chapter three, we present a physics-constrained deep learning method to discover invariant NNMs, encapsulating the spatiotemporal dynamics of fluid flows with strong nonlinearity. The NNM-physics-constrained convolutional autoencoder (NNM-CNN-AE) captures the nonlinear modal transformation, NNMs, and enables reduced-order reconstruction and long-term future-state prediction of flow fields. Testing on various flow regimes around a cylinder demonstrates that NNMs reveal spatiotemporal dynamics effectively and outperform linear proper orthogonal decomposition (POD) and Koopman-based methods in accuracy and efficiency.

Moreover, in chapter four, we address the challenge of modeling parameterized nonlinear systems without complete knowledge of underlying physics. The dynamics-embedded conditional generative adversarial network (Dyn-cGAN) identifies temporal dynamics and their dependence on system parameters, facilitating long-term prediction of responses. Numerical studies on various parameterized nonlinear systems illustrate the effectiveness of the Dyn-cGAN for predictive modeling, particularly in capturing chaotic behaviors.