Jonathan H. Tu

Compressive Sensing and Low-Rank Libraries for Classification of Bifurcation Regimes in Nonlinear Dynamical Systems

We show that for complex nonlinear systems, model reduction and compressive sensing strategies can be combined to great advantage for classifying, projecting, and reconstructing the relevant low-dimensional dynamics.

Spectral analysis of fluid flows using sub-Nyquist-rate PIV data

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An improved algorithm for balanced POD through an analytic treatment of impulse response tails

-based dimensionality reduction methods such as the proper orthogonal decomposition are used to construct separate modal libraries and Galerkin models based on data from a number of bifurcation regimes. These libraries are then concatenated into an overcomplete library, and

Algorithm 945: modred—A Parallelized Model Reduction Library

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Koopman spectral analysis of separated flow over a finite-thickness flat plate with elliptical leading edge

-sparse representation in this library from a few noisy measurements results in correct identification of the bifurcation regime. This technique provides an objective and general framework for classifying the bifurcation parameters and, therefore, the underlying dynamics and stability. After classifying the bifurcation regime, it is possible to employ a low-dimensional Galerkin model, only on modes relevant to that bifurcation value. These methods are demonstrated on the complex Ginzburg--Landau equation usin...

Control of a canonical separated flow

Abstract Spectral methods are ubiquitous in the analysis of dynamically evolving fluid flows. However, tools like Fourier transformation and dynamic mode decomposition (DMD) require data that satisfy the Nyquist–Shannon sampling criterion. In many fluid flow experiments, such data are impossible to acquire. We propose a new approach that combines ideas from DMD and compressed sensing to accommodate sub-Nyquist-rate sampling. Given a vector-valued signal, we take measurements randomly in time (at a sub-Nyquist rate) and project the data onto a low-dimensional subspace. We then use compressed sensing to identify the dominant frequencies in the signal and their corresponding modes. We demonstrate this method using two examples, analyzing both an artificially constructed dataset and particle image velocimetry data from the flow past a cylinder. In each case, our method correctly identifies the characteristic frequencies and oscillatory modes dominating the signal, proving it to be a capable tool for spectral analysis using sub-Nyquist-rate sampling.

Integration of non-time-resolved PIV and time-resolved velocity point sensors for dynamic estimation of time-resolved velocity fields

We present a modification of the balanced proper orthogonal decomposition (balanced POD) algorithm for systems with simple impulse response tails. In this new method, we use dynamic mode decomposition (DMD) to estimate the slowly decaying eigenvectors that dominate the long-time behavior of the direct and adjoint impulse responses. This is done using a new, low-memory variant of the DMD algorithm, appropriate for large datasets. We then formulate analytic expressions for the contribution of these eigenvectors to the controllability and observability Gramians. These contributions can be accounted for in the balanced POD algorithm by simply appending the impulse response snapshot matrices (direct and adjoint, respectively) with particular linear combinations of the slow eigenvectors. Aside from these additions to the snapshot matrices, the algorithm remains unchanged. By treating the tails analytically, we eliminate the need to run long impulse response simulations, lowering storage requirements and speeding up ensuing computations. To demonstrate its effectiveness, we apply this method to two examples: the linearized, complex Ginzburg-Landau equation, and the two-dimensional fluid flow past a cylinder. As expected, reduced-order models computed using an analytic tail match or exceed the accuracy of those computed using the standard balanced POD procedure, at a fraction of the cost.

Pulse Formation, Harmonic Mode-Locking, and Stability in Actively Mode-Locked Laser Cavities

We describe a new parallelized Python library for model reduction, modal analysis, and system identification of large systems and datasets. Our library, called modred, handles a wide range of problems and any data format. The modred library contains implementations of the Proper Orthogonal Decomposition (POD), balanced POD (BPOD) Petrov-Galerkin projection, and a more efficient variant of the Dynamic Mode Decomposition (DMD). The library contains two implementations of these algorithms, each with its own advantages. One is for smaller and simpler datasets, requires minimal knowledge to use, and follows a common matrix-based formulation. The second, for larger and more complicated datasets, preserves the abstraction of vectors as elements of a vector space and, as a result, allows the library to work with arbitrary data formats and eases distributed memory parallelization. We also include implementations of the Eigensystem Realization Algorithm (ERA), and Observer/Kalman Filter Identification (OKID). These methods are typically not computationally demanding and are not parallelized. The library is designed to be easy to use, with an object-oriented design, and includes comprehensive automated tests. In almost all cases, parallelization is done internally so that scripts that use the parallelized classes can be run in serial or in parallel without any modifications.

Variants of Dynamic Mode Decomposition: Boundary Condition, Koopman, and Fourier Analyses

The separated flow over an airfoil is characterized by up to three distinct natural frequencies: those of the shear layer, separation bubble, and wake. Previous work has shown that open-loop forcing at sub- and super- harmonics of these frequencies can be especially effective in controlling the extent of the separation bubble. Unfortunately, an understanding of the mechanisms driving this behavior is far from complete. In this work, we investigate the interactions between the shear layer and wake using a combination of direct numerical simulations and spectral analysis. We simulate the forced and unforced flows over a finite-thickness flat plate using an immersed boundary method. Spectral analysis of the resulting dynamics is performed using the Koopman operator, a linear operator applicable even to nonlinear systems, as well as traditional signal processing techniques. Using these two approaches, we identify pertinent flow structures based on their frequency content and posit the nature of their interactions.

Integration of non-time-resolved PIV and time-resolved velocity point sensors for dynamic estimation of velocity fields

A stalled airfoil can exhibit up to three natural frequencies associated with the separated flow: that of the shear layer, the separation bubble, and the wake. This work investigates these flow phenomena using a simplified canonical setup and targets their frequencies with zero-net mass-flux (ZNMF) actuation in order to effectively and efficiently reduce the extent of the separation. First, boundary layer separation is created on a flat plate model, devoid of curvature that would otherwise include effects particular to the type of aerodynamic body, by imposing an adverse pressure gradient via a ZNMF suction/blowing boundary condition on the tunnel ceiling. At a chord Reynolds number of 10, the nominal two-dimensional characteristics of the flow are verified. The uncontrolled flow is characterized, including identification of the key flow frequency content, prior to strategically targeting this content with unsteady actuation. The identified natural frequencies of the shear layer and wake are then targeted via open-loop ZNMF sinusoidal and burst-modulated (BM) forcing. The results clearly indicate the ability to reattach the separated flow using significantly reduced Cμ values via BM forcing compared to sinusoidal forcing at the actuator resonance frequency.