Bulletin of the American Physical Society
73rd Annual Meeting of the APS Division of Fluid Dynamics
Volume 65, Number 13
Sunday–Tuesday, November 22–24, 2020; Virtual, CT (Chicago time)
Session K09: Nonlinear Dynamics: Model Reduction (8:45am  9:30am CST)Interactive On Demand

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K09.00001: Datadriven Modeling of Detonation Wave Interactions in Rotating Detonation Engines Ariana Mendible, James Koch, Henning Lange, Steven Brunton, Nathan Kutz Direct observation of a Rotating Detonation Engine (RDE) combustion chamber has enabled the extraction of the kinematics of its detonation waves. The resulting combustion fronts are composed of co and counterrotating coherent traveling shock waves whose nonlinear dynamics are rife with modelocking behavior, bifurcations, and instabilities which are not well understood. Computational fluid dynamics simulations are ubiquitous in the endeavor to characterize the dynamics of RDEs, however, they prove to be prohibitively expensive when considering multiple engine geometries or operating conditions. Reduced order models (ROMs) are preferred to direct calculations because they exploit lowrank structure in the data to minimize computational cost and allow for rapid parameterized studies. ROMs are inherently inhibited by translational invariances such as the traveling waves present in RDEs. In this work, we overcome these roadblocks by using machine learning to discover moving coordinate frames into which the data is shifted. This allows for the application of traditional dimensionality reduction techniques. We explore a suite of datadriven models, ranging from simple representations to neural networks, describing the complex dynamics of the shock waves in the RDE. [Preview Abstract] 

K09.00002: Datadriven nonlinear aeroelastic models of morphing wings for control Urban Fasel, Nicola Fonzi, Steven L. Brunton Accurate and efficient aeroelastic models are important for enabling the optimization and control of morphing wings, which are characterized by highlycoupled and nonlinear interactions between the aerodynamic and structural dynamics. In this work, we leverage emerging datadriven modeling techniques to develop highly accurate and tractable reducedorder aeroelastic models that are valid over a wide range of operating conditions and are suitable for control. In particular, we develop two extensions to the dynamic mode decomposition with control (DMDc) algorithm to make it suitable for aeroelastic systems: 1) we introduce a formulation to handle algebraic equations, and 2) we develop an interpolation scheme to connect several linear DMDc models developed in different operating regimes. Thus, the innovation lies in accurately modeling the nonlinearities of the coupled aerostructural dynamics over multiple operating regimes. We demonstrate this approach on a highfidelity model of an airborne wind energy (AWE) system, although the methods are generally applicable to any highly coupled aeroelastic system. Our proposed modeling framework results in realtime prediction and we demonstrate the enhanced model performance for model predictive control. Thus, the proposed architecture may help enable the widespread adoption of nextgeneration morphing wing technologies. [Preview Abstract] 

K09.00003: Learning dominant physical processes in complex flows with datadriven balance models Jared Callaham, J. Nathan Kutz, Bingni Brunton, Steven Brunton Theoretical analysis in fluid mechanics has long relied on judiciously approximating the full flow physics as a balance between a few dominant processes. However, this traditional approach typically only applies in asymptotic regimes where there is a strict separation of scales. We automate and generalize this approach to nonasymptotic regimes by introducing the idea of an equation space, in which different local balances appear as distinct subspace clusters. Unsupervised learning can then automatically identify regions where groups of terms may be neglected. We show that our datadriven balance models successfully delineate dominant balance physics in a broad range of fluid flows, including a laminar bluff body wake, a boundary layer in transition to turbulence, and surface currents in the Gulf of Mexico. [Preview Abstract] 

K09.00004: Robust Principal Component Analysis for Modal Decomposition of Corrupt Fluid Flows Isabel Scherl, Benjamin Strom, Jessica K. Shang, Owen Williams, Brian L. Polagye, Steven L. Brunton Modal analysis techniques have been used to identify patterns and structures in a variety of fluid applications. However, experimentally acquired flow fields may be corrupted with incorrect and missing entries, which will degrade subsequent modal analyses. Here we explore how robust principal component analysis (RPCA) can be used to leverage global coherent structures to identify and replace spurious data points. RPCA decomposes a data matrix into a sparse component and lowrank matrix of coherent structure. We explore RPCA on a range of fluid simulations and experiments of varying complexity and assess how accurately the low rank structure in the data is recovered.In all cases, we find that RPCA extracts dominant fluid coherent structures and identifies and fills in incorrect or missing measurements. The performance is particularly striking when flow fields are analyzed using dynamic mode decomposition, which is sensitive to noise and outliers. [Preview Abstract] 

K09.00005: Sequential sampling with heteroscedastic surrogate model to quantify extreme response statistics Xianliang Gong, Yulin Pan We consider a dynamical system with two sources of uncertainties: (1) parameterized input with known probability distribution, and (2) stochastic inputtoresponse (ItR) map. Our purpose is to efficiently quantify the extreme response statistics when the ItR map is expensive to compute (so a full MonteCarlo approach is not affordable). This problem setup arises often in physics and engineering, such as weather forecasting and ship motion in irregular wave fields (where the stochasticity in ItR can come from the uncertain subgrid processes and initial conditions in simulation for each input). Our approach in essence leverages sequential sampling in the input parameter space, and accounts for the stochastic ItR map through the heteroscedastic Gaussian process regression (HGPR). A sequential sampling (i.e., nextbest sampling) criterion is developed which minimizes the required number of samples to achieve accurate resolution of the extreme response statistics. We demonstrate the effectiveness of the current approach with multiple examples including the ship motion response problem. [Preview Abstract] 

K09.00006: Phasebased analysis of synchronization between laminar cylinder wake and external harmonic actuations Mohammad Amin Khodkar, Kunihiko Taira We leverage phasereduction theory to explore the synchronization properties of the twodimensional periodic flow over a circular cylinder at Reynolds number of 100. Towards this end, a direct method based on applying weak impulse perturbations at various locations in the near wake and at several times over a period is adopted. The phase response of the wake flow to these impulses reveals how its dynamics can be impacted by external periodic actuations, through enabling the development of a onedimensional and linear phasebased model with respect to the limit cycle, in place of the full highdimensional and nonlinear dynamics of the wake. Comparison between the results of the current work and those provided by the Koopman or adjointbased analysis of the flow illustrates the commonalities and differences between these methods, while highlighting the characteristic features and capabilities of the phasereduction analysis. The excellent predictions of the present model with regard to the synchronization properties of the flow under consideration holds promise for its application to complex engineering problems such as structural vibrations and biological swimmers and flyers. [Preview Abstract] 

K09.00007: Network broadcast mode analysis and control with respect to timevarying base flow1 ChiAn Yeh, Muralikrishnan Gopalakrishnan Meena, Kunihiko Taira We present a networkbased modal analysis that identifies the key dynamical paths along which perturbations amplify in highly unsteady flows. This analysis is built upon the Katz centrality to reveal the flow structures that can effectively spread perturbations over the timeevolving network of vortical elements. Motivated by the resolvent form of the Katz function, we take the singular value decomposition of the resulting communicability matrix, complementing resolvent analysis for fluid flows. The rightsingular vector, referred to as the broadcast mode, gives insights into the sensitive regions where introduced perturbations can be effectively spread over the entire fluidflow network as it evolves over time. We apply the developed formulation to the example of twodimensional decaying isotropic turbulence. The broadcast mode identifies vortex dipoles as the important structures in amplifying perturbations. By perturbing the flow with the broadcast mode, we demonstrate the effective modification of turbulent flows. The current networkinspired work presents a novel use of network analysis to guide flow control efforts for timevarying base flows. [Preview Abstract] 

K09.00008: Wavelet adaptive POD for large scale flow data Philipp Krah, Thomas Engels, Kai Schneider, Julius Reiss The proper orthogonal decomposition (POD) is a powerful classical tool in fluid mechanics used for instance for model reduction and extraction of coherent flow features. However its applicability to high resolution 3D DNS data is limited due to its computational complexity. Here we propose a waveletbased adaptive POD, called wPOD, which overcomes this limitation. The size of the analyzed data is reduced by exploiting the compression properties of wavelets with error control, which yields a sparse flow representation. Numerical analysis shows that wavelet compression and POD truncation errors can be balanced and massive 3d high resolution data sets can thus be efficiently handled. A validation of the method will be given for 2D wake flow data. Examples will then be presented for 3D high resolution DNS data of flapping insect flight in turbulence. A comparison with the randomized singular value decomposition illustrates the efficiency and the precision of the wPOD method. [Preview Abstract] 

K09.00009: Projection and Tree Based Reduced Order Modeling for Vortex Particle Simulations. Steven Rodriguez, Athanasios Iliopoulos, Steven Brunton, Kevin Carlberg, John Michopoulos, John Steuben Vortex particle methods are ubiquitous in modeling vorticity transport phenomena. Example applications include the modeling of a helicopter rotor wake, or wakebody interactions in a school of fish. Unfortunately, these vortex particle methods exhibit poor quadratic $O(N^{2})$ operationcount complexity (OCC), with respect to the number of $N$ particles in the domain. Acceleration techniques, such as the fastmultipole method or other treemethods, can be used to reduce the OCC. However, these techniques have at best reduced computations to an \textit{Ndependent} linear OCC, i.e. $O(N)$. The presented work addresses the Ndependent OCC bottleneck by introducing a framework that combines hierarchical decomposition and projectionbased hyperreduction to enable \textit{Nindependent} OCC. Specifically, the presented framework combines the BarnesHut tree method with GNAT hyperreduction to reduce the pairwise interactions of an Nbody problem. The presented method will be showcased on the BiotSavart kernel to demonstrate fast computations of the induced velocity field for parametric fluiddynamic example problems. [Preview Abstract] 

K09.00010: A hierarchy structure of local Koopman spectra Wei Zhang, Mingjun Wei Local Koopman spectrum is studied for its role in resolving dynamics of a nonlinear system. For typical linear systems, local Koopman spectra and eigenspace are described by linear theories; for nonlinear systems, the proliferation rule is observed in its recursive application in nonlinear observables. A hierarchy structure of Koopman eigenspace is therefore introduced to depict the nonlinear dynamics in general. With the dynamics being decoupled to base and perturbation components, the perturbation is further separated to linear and nonlinear parts. In the hierarchy, the linear part follows linear spectral theories, and the nonlinear part is defined by linear spectra through recursive proliferation. Local Koopman spectra and eigenfunctions evolve continuously in the whole manifold and are derived from operator perturbation theory. The above analyses were applied first to LTI systems, and then to the nonlinear transition of the flow passing a fixed cylinder and its final periodic state. The triadchain and the lattice distribution of Koopman spectra are observed numerically to confirm the hierarchy structure of Koopman spectra and the critical role of proliferation rule being involved. [Preview Abstract] 

K09.00011: Leastorder model for the transient dynamics of the fluidic pinball Nan Deng, Luc R. Pastur, Bernd R. Noack, Marek Morzynski We propose a leastorder meanfield model for a flow system undergoing two successive supercritical bifurcations. The fluidic pinball, an incompressible twodimensional flow crossing three equidistantly spaced cylinders, is numerically investigated using Direct Numerical Simulation. Two generic bifurcations in fluid mechanics are observed: the primary Hopf bifurcation leads to a statistically symmetric vortex shedding and the following pitchfork bifurcation breaks the symmetry at higher Reynolds number. Interestingly, this symmetrybreaking instability works on the steady solution simultaneously, illustrated by the global stability analysis and Floquet analysis. The elementary degrees of freedom are identified under meanfield considerations exploiting the symmetry/asymmetry of the base flow and the fluctuation. An easily interpretable fivedimensional Galerkin model compatible with the quadratic nonlinearities of the NavierStokes equations is derived, which can reproduce the main features of bifurcating dynamics and the transient behavior to the asymptotic regime. This generalized meanfield Galerkin methodology is considered to be applicable to other transition scenarios and nonlinear modelbased control. [Preview Abstract] 

K09.00012: Towards equationfree resolvent analysis Steven Brunton, Peter Baddoo, Benjamin Herrmann, Beverley McKeon As an equationbased method, resolvent analysis requires knowledge of the exact governing equations of the system so that the resolvent operator can be computed. In this presentation, we explore whether resolvent analysis can be performed purely from data. We base our approach on a nonlinear version of dynamic mode decomposition (DMD) that allows us to approximate the underlying nonlinear operator. Since the system is typically of very high dimension, we leverage techniques from machine learning to project the data onto the principal components of a high (or infinite) dimensional latent space. This approach enables us to disambiguate the linear and nonlinear parts of the operator in the original physical domain. We then establish a connection to the resolvent operator and, finally, validate our approach on a range of highdimensional problems. [Preview Abstract] 

K09.00013: When deep learning meets ergodic theory Michele Alessandro Bucci, Onofrio Semeraro, Sergio Chibbaro, Alexandre Allauzen, Lionel Mathelin The reliable prediction of the temporal behavior of complex systems is required in numerous fields, including fluid mechanics. This strong interest is inherently connected with modeling issues: often, the governing equations describing the physics of the system under consideration are not accessible or, when known, their solution might require a computational time incompatible with the prediction time constraints. In this view, a datadriven approach is to approximate the system at hand in a generic functional format and inform it from available observations. Numerous successful examples are already available based on deep Neural Networks. Here, we consider LongShortTerm Memory neural networks and thoroughly investigate the impact of the training set on the longterm prediction. Leveraging metrics and measures from ergodic theory, we analyze the amount of data sufficient for a priori guaranteeing a faithful model of the real system. In particular, we will show how an informed design of the training set, based on symmetries of the system and the topology of underlying attractor, significantly improves the resulting models, opening up avenues of research within the context of active learning. Our developments will be illustrated on the Lorenz’63 model and shear flow models. [Preview Abstract] 

K09.00014: Clusterbased network model Hao LI, Daniel Fernex, Richard Semaan, Jianguo TAN, Marek Morzy\’nski, Bernd R. Noack We propose an automatable datadriven methodology for robust nonlinear reducedorder modelling from timeresolved snapshot data. In the kinematical coarsegraining, the snapshots are clustered into few centroids representable for the whole ensemble. The dynamics is conceptualized as a directed network, where the centroids represent nodes and the directed edges denote possible finitetime transitions. The transition probabilities and times are inferred from the snapshot data. The resulting clusterbased network model constitutes a deterministicstochastic greybox model resolving the coherentstructure evolution. This model is motivated by limitcycle dynamics, illustrated for the chaotic Lorenz attractor and successfully demonstrated for the laminar twodimensional mixing layer featuring KelvinHelmholtz vortices and vortex pairing, and for an actuated turbulent boundary layer with complex dynamics. Clusterbased network modelling opens a promising new avenue with unique advantages over other modelorder reductions based on clustering or proper orthogonal decomposition. [Preview Abstract] 

K09.00015: Datadriven sensor selection method using ADMM for a largescale problem Takayuki Nagata, Nonomura Taku, Kumi Nakai, Keigo Yamada, Yuji Saito, Shunsuke Ono The present study proposes a sensor selection method based on the sparsitypromoting framework with the Aoptimality criterion and the alternating direction method of multipliers algorithm. The performance of the proposed method was evaluated with a random sensor problem and compared with the previously proposed methods such as the greedy method and the convex relaxation. The performance of the proposed method was better than the existing method in terms of the trace of the inverse of the Fisher information matrix. The computational complexity of the proposed method is the first order of the size of the problem ($n)$ and the square order of the number of latent state variables ($r)$. In the case of the convex approximation, the computational complexity is cubic orders of the size of the problem. The Considered problem in the present study is $r$~\textless \textless ~$n$, and thus the computational cost of the proposed method is quite smaller than that of the convex relaxation. The proposed method was applied to the datadriven sparsesensorselection problem. A data set adopted is the NOAA OISST V2 mean sea surface temperature set. At the number of sensors larger than that of the latent state variables, the proposed method showed better performance compared to previously proposed methods in terms of the trace of the inverse of the Fisher information matrix and reconstruction error. [Preview Abstract] 

K09.00016: Effect of Objective Function on Sparse Sensor Placement using Greedy Method Kumi Nakai, Keigo Yamada, Takayuki Nagata, Yuji Saito, Taku Nonomura In the present study, the objective functions based on Doptimal, Aoptimal, and Eoptimal criteria of optimal design are adopted to the datadriven sparse sensor selection using the greedy method. The D, A, and Eoptimal design maximizes the determinant, minimizes the trace of inverse, and maximizes the minimum eigenvalue of the Fisher information matrix, respectively. The greedy methods based on D, A, and Eoptimality are applied to a random sensor problem, and computational results are compared. In terms of the determinant and trace of the inverse, the sensors selected by the Doptimality objective function works better than those by A and Eoptimality. On the other hand, in terms of the minimum eigenvalue, those by Aoptimality works the best and those by Eoptimality works better than those by Doptimality. Furthermore, the climate datasets of the National Oceanic and Atmospheric Administration (NOAA) are reconstructed using the sensors selected by the D, A, and Eoptimalitybased greedy methods. The results indicate that the greedy method based on Doptimality is the most suitable for high accurate reconstruction with low computational cost in the case of training datasets are the same as test datasets. [Preview Abstract] 

K09.00017: Assessment of hybrid datadriven models to predict unsteady flows Rachit Gupta, Sandeep Reddy Bukka, Rajeev Jaiman This work systematically assesses two hybrid datadriven reducedorder models for predicting unsteady wake dynamics for single and sidebyside cylinders. These models rely on recurrent neural networks (RNNs) to evolve lowdimensional unsteady flow states. The first model, termed PODRNN, projects the highfidelity NavierStokes data to a lowdimensional subspace via proper orthogonal decomposition (POD). The timedependent coefficients in the POD subspace are propagated by recurrent net (closedloop/encoderdecoder updates) and mapped to a highdimensional state via the mean flow field and POD basis vectors. The second model, referred as convolution recurrent autoencoder network (CRAN), employs convolutional neural networks (CNN) (instead of POD), as layers of linear kernels with nonlinear activations, to extract lowdimensional features from flow snapshots. The flattened features are advanced using a recurrent (closedloop manner) net and upsampled gradually to highdimensional snapshots. For flow past a cylinder, the predictive performance of both models is remarkable, with CRAN being a bit overkill. However, CRAN outperforms PODRNN for longer time predictions of bistable flow past sidebyside cylinders. [Preview Abstract] 

K09.00018: Learning fluid flow physics from noisy, incomplete, experimental data Logan Kageorge, Patrick Reinbold, Michael Schatz, Roman Grigoriev Purely datadriven methods have shown a lot of promise in identifying models of simple, lowdimensional systems from data which have a low level of noise and provide a complete description of the system state. However, they fall apart for data that is highdimensional, noisy, or incomplete, which is common in fluid dynamics. We show that this challenge can be addressed by augmenting the datadriven approach with a few general physical constraints and using a weak formulation of the model. To illustrate this, we construct a quantitative twodimensional model of a weakly turbulent flow in a thin layer of electrolyte driven by Lorentz force from PIV data on a coarse spatiotemporal grid. Our hybrid approach also allows reconstruction of the latent variables that cannot be measured directly, e.g., pressure and forcing field. [Preview Abstract] 

K09.00019: AdaptiveRobust Dynamic Mode Decomposition (DMD) for ReducedOrder Modeling and Control Saleh Nabi, Aniketh Kalur, Mouhacine Benosman We develop a datadriven reducedorder modeling framework that adapts to changing flow regimes. Such a framework ensures the reducedorder model (ROM) corrects itself to account for changing parameters. Generally, ROMs developed from the dataassimilation are built on specific flow parameters for which the data is available. However, it is ubiquitous that the flow regime is often changing in systems such as HVAC applications, wind farms, etc. Therefore, ROMs need to account for parameter variations to be useful in an “offdesign” setting. In this work, we develop a datadriven ROM at a nominal parameter using Dynamic Mode Decomposition (DMD), following which the ROM is appended with a correction/closure term that adapts itself in an online manner to correct for varying parameters. First, we show that the correction term robustly stabilizes the system by guaranteeing stability. Secondly, we use an extremum seeking optimization approach to update the correction term parameters to adapt to the aforementioned variations. Lastly, we discuss the results of using our proposed framework on the viscous Burgers equation. We show that our framework can adapt and account for parametric uncertainties by selfcorrecting even though the ROM is developed in a flow regime ``very far'' from the gr [Preview Abstract] 

K09.00020: Probabilistic clusterbased feedback control of fluid flow dynamics Vedasri Godavarthi, ChiAn Yeh, Eurika Kaiser, Kunihiko Taira We develop a clusterbased feedback control strategy to modify the evolution of flow states to a desired dynamics using limited temporal measurements capturing the flow’s attractor. The flow state is encoded into a lowdimensional feature space, which is partitioned into a small number of clusters and where each flow state is represented by a cluster probability vector. The evolution of the cluster probability vector is given based on the transition probability matrix, which encodes the transitions among the clusters. We formulate an LQR problem to control the cluster probability vector to a desired distribution. The control gain matrix is translated to an optimal feedback control law in the physical space using an estimated scaling function. We first demonstrate this probabilistic control approach on two canonical oscillators: a Lorenz63 system with bistability and a coupled FitzHugh Nagumo oscillator system that exhibit extreme events. Using DNS, we also employ this control strategy to reduce the pressure fluctuations over a 2D cavity using limited sensor measurements on the cavity wall. The present approach is developed for realtime characterization and control of complex flow dynamics. [Preview Abstract] 

K09.00021: Neural Network approach to reduced order modeling of multiphase flows Cristina P. Martin Linares, Tom Bertalan, Jiacai Lu, Seungjoon Lee, Yannis Kevrekidis, Gretar Tryggvason We explore the use of Neural Networks (NN) to learn black as well as grey box models of the Partial Differential Equations (PDE) that govern multiphase flows in a 2Dimensional (2D) vertical channel. The data is generated using Direct Numerical Simulations (DNS). The covariance method is used to perform Proper Orthogonal Decomposition (POD) on the velocity and void fraction to filter the data so we can learn an effective PDE. The selected POD modes are further reduced through an autoencoder. The selected minimum number of nonlinear projections have a onetoone correspondence with the first few POD modes while reducing the loss function. The POD modes are used to evolve the solution in time using NN. We also use a NN to learn the functional form of the PDE and use the learned PDE to predict the dynamics. The closure terms in the averaged multiphase flow equations are predicted using NN and the predicted PDE is used to evolve in time the velocity and the void fraction, in another method. The developed models are used to predict the dynamics of flows with different initial and boundary conditions. [Preview Abstract] 
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