Reduced order modeling of convection-dominated flows, dimensionality reduction and stabilization

Mojgani, Rambod

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https://hdl.handle.net/2142/108494 Copy

Description

Title

Reduced order modeling of convection-dominated flows, dimensionality reduction and stabilization

Author(s)

Mojgani, Rambod

Issue Date

2020-07-16

Director of Research (if dissertation) or Advisor (if thesis)

Balajewicz, Maciej

Doctoral Committee Chair(s)

Bodony, Daniel J

Committee Member(s)

• Panesi, Marco
• Goza, Andres
• Gazzola, Mattia

Department of Study

Aerospace Engineering

Discipline

Aerospace Engineering

Degree Granting Institution

University of Illinois at Urbana-Champaign

Degree Name

Ph.D.

Degree Level

Dissertation

Keyword(s)

• reduced order model
• stabilization, auto-encoder
• time-varying systems
• convection
• convection dominated flows
• hyperbolic partial differential equation

Abstract

We present methodologies for reduced order modeling of convection dominated flows. Accordingly, three main problems are addressed. Firstly, an optimal manifold is realized to enhance reducibility of convection dominated flows. We design a low-rank auto-encoder to specifically reduce the dimensionality of solution arising from convection-dominated nonlinear physical systems. Although existing nonlinear manifold learning methods seem to be compelling tools to reduce the dimensionality of data characterized by large Kolmogorov n-width, they typically lack a straightforward mapping from the latent space to the high-dimensional physical space. Also, considering that the latent variables are often hard to interpret, many of these methods are dismissed in the reduced order modeling of dynamical systems governed by partial differential equations (PDEs). This deficiency is of importance to the extent that linear methods, such as principle component analysis (PCA) and Koopman operators, are still prevalent. Accordingly, we propose an interpretable nonlinear dimensionality reduction algorithm. An unsupervised learning problem is constructed that learns a diffeomorphic spatio-temporal grid which registers the output sequence of the PDEs on a non-uniform time-varying grid. The Kolmogorov n-width of the mapped data on the learned grid is minimized. Secondly, the reduced order models are constructed on the realized manifolds. We project the high fidelity models on the learned manifold, leading to a time-varying system of equations. Moreover, as a data-driven model free architecture, recurrent neural networks on the learned manifold are trained, showing versatility of the proposed framework. Finally, a stabilization method is developed to maintain stability and accuracy of the projection based ROMs on the learned manifold a posteriori. We extend the eigenvalue reassignment method of stabilization of linear time-invariant ROMs, to the more general case of linear time-varying systems. Through a post-processing step, the ROMs are controlled using a constrained nonlinear lease-square minimization problem. The controller and the input signals are defined at the algebraic level, using left and right singular vectors of the reduced system matrices. The proposed stabilization method is general and applicable to a large variety of linear time-varying ROMs.

Graduation Semester

2020-08

Type of Resource

Thesis

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http://hdl.handle.net/2142/108494

Copyright and License Information

Copyright 2020 Rambod Mojgani

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