Higher Order Dynamic Mode Decomposition Research Articles

Higher-order dynamic mode decomposition on-the-fly: A low-order algorithm for complex fluid flows

This article presents a new method to identify the main patterns describing the flow motion in complex flows. The algorithm is an extension of the higher-order dynamic mode decomposition (HODMD), which compresses the snapshots from the analysed database and progressively updates new compressed snapshots on-the-fly, so it is denoted as HODMD on-the-fly (HODMD-of). This algorithm can be applied in parallel to the numerical simulations (or experiments), and it exhibits two main advantages over offline algorithms: (i) it automatically selects on-the-fly the number of necessary snapshots from the database to identify the relevant dynamics; and (ii) it can be used from the beginning of a numerical simulation (or experiment), since it uses a sliding-window to automatically select, also on-the-fly, the suitable interval to perform the data analysis, i.e. it automatically identifies and discards the transient dynamics. The HODMD-of algorithm is suitable to build reduced order models, which have a much lower computational cost than the original simulation. The performance of the method has been tested in three different cases: the axi-symmetric synthetic jet, the three-dimensional wake of a circular cylinder and the turbulent wake behind a wall-mounted square cylinder. The obtained speed-up factors are around 7 with respect to HODMD; this value depends on the simulation and the configuration of the hyperparameters. HODMD-of also provides a significant reduction of the memory requirements, between 40−80% amongst the two- and three-dimensional cases studied in this paper.

A novel data-driven method for the analysis and reconstruction of cardiac cine MRI

Cardiac cine magnetic resonance imaging (MRI) can be considered the optimal criterion for measuring cardiac function. This imaging technique can provide us with detailed information about cardiac structure, tissue composition and even blood flow, which makes it highly used in medical science. But due to the image time acquisition and several other factors the MRI sequences can easily get corrupted, causing radiologists to misdiagnose 40 million people worldwide each and every single year. Hence, the urge to decrease these numbers, researchers from different fields have been introducing novel tools and methods in the medical field. Aiming to the same target, we consider in this work the application of the higher order dynamic mode decomposition (HODMD) technique. The HODMD algorithm is a linear method, which was originally introduced in the fluid dynamics domain, for the analysis of complex systems. Nevertheless, the proposed method has extended its applicability to numerous domains, including medicine. In this work, HODMD in used to analyze sets of MR images of a heart, with the ultimate goal of identifying the main patterns and frequencies driving the heart dynamics. Furthermore, a novel interpolation algorithm based on singular value decomposition combined with HODMD is introduced, providing a three-dimensional reconstruction of the heart. This algorithm is applied (i) to reconstruct corrupted or missing images, and (ii) to build a reduced order model of the heart dynamics.

Higher order dynamic mode decomposition of an experimental trailing vortex

The decay of trailing vortices is a fundamental problem in fluid mechanics and constitutes the basis of control applications that intend to alleviate the wake hazard. In order to progress, we use the recently developed modal-decomposition technique to identify the governing dynamics in an experimental trailing vortex. A particular emphasis is on the difficulty and usefulness of applying such tools to noisy experimental data. We conducted a water-tunnel experiment at a chord-based Reynolds number Re=4×104 using stereoscopic particle image velocimetry measurements over a downstream range of 36 chords. The downstream evolution of the maximum of vorticity suggests that the whole wake can be partitioned into three consecutive regimes. A higher-order dynamic mode decomposition of the streamwise vorticity in each such part of the wake shows that the decay is well approximated by at most three modes. Additionally, our study provides evidence for the existence of several instabilities after the vortex roll up beyond about 6.5 chords.

Analyzing Slightly Inclined Cylindrical Binary Fluid Convection via Higher Order Dynamic Mode Decomposition

Analyzing Slightly Inclined Cylindrical Binary Fluid Convection via Higher Order Dynamic Mode Decomposition

Crack Localization in Operating Rotors Based on Multivariate Higher Order Dynamic Mode Decomposition.

A novel output-only crack localization method is proposed for operating rotors based on an enhanced higher-order dynamic mode decomposition (HODMD), in which the nonlinear breathing crack-induced super-harmonic characteristic components from multiple vibration measurement points are simultaneously extracted to compose the corresponding super-harmonic transmissibility damage indexes. Firstly, the theoretical background of the HODMD is briefly reviewed. Secondly, the proposed crack localization method is dedicated which improving the HODMD for multivariate signals by casting the total least square method into standard HODMD and adaptively selecting the order parameter of Koopman approximation by optimizing the super-harmonic frequency vector. In addition, the super-harmonic characteristic components are evaluated and harnessed to derive the damage index based on super-harmonic transmissibility and fractal dimension. Finally, the proposed method is investigated and demonstrated by numerical simulations and experiments. Both numerical and experimental results show that the proposed method is powerful in realizing multi-crack localization for running rotors accurately and robustly in the case of no baseline information on intact rotors. Moreover, the interferences from commonly existing steps and misalignment can also be eliminated.

An Adaptive Data-Driven Reduced Order Model Based on Higher Order Dynamic Mode Decomposition

An Adaptive Data-Driven Reduced Order Model Based on Higher Order Dynamic Mode Decomposition

On the generation and destruction mechanisms of arch vortices in urban fluid flows

This study uses higher-order dynamic mode decomposition to analyze a high-fidelity database of the turbulent flow in an urban environment consisting of two buildings separated by a certain distance. We recognize the characteristics of the well-known arch vortex forming on the leeward side of the first building and document this vortex's generation and destruction mechanisms based on the resulting temporal modes. We show that the arch vortex plays a prominent role in the dispersion of pollutants in urban environments, where its generation leads to an increase in their concentration; therefore, the reported mechanisms are of extreme importance for urban sustainability.

Data-driven analysis of oscillations in Hall thruster simulations

A data-driven modal analysis of plasma oscillations in a stationary plasma thruster-100-like Hall thruster in the 1–120 kHz range is presented. Data are generated by a two-dimensional (axial-radial) hybrid particle-in-cell/fluid simulation code. While proper orthogonal decomposition is unable to successfully uncouple the different dynamics, higher order dynamic mode decomposition (HODMD) cleanly isolates the breathing and ion transit time (ITT) modes. Indeed, the computed HODMD components can be clustered into two distinct groups, enabling the separate reconstruction of the dynamics of the two oscillation modes. It is also shown that each plasma variable exhibits a different behavior in each cluster. The breathing oscillations have a global or standing-wave character for the plasma density, neutral density and ion current density inside the thruster, and an axially progressive-wave structure for the electron temperature and plasma potential in the near plume. The ITT mode shows a progressive-wave structure for all variables, except for the neutral density whose amplitude is negligible. This work serves as an illustration of the applicability of data-driven techniques like HODMD to the analysis of plasma flows relevant to electric propulsion.

Higher order dynamic mode decomposition: From fluid dynamics to heart disease analysis

In this work, we study in detail the performance of Higher Order Dynamic Mode Decomposition (HODMD) technique when applied to echocardiography images. HODMD is a data-driven method generally used in fluid dynamics and in the analysis of complex non-linear dynamical systems modeling several complex industrial applications. In this paper we apply HODMD, for the first time to the authors knowledge, for patterns recognition in echocardiography, specifically, echocardiography data taken from several mice, either in healthy conditions or afflicted by different cardiac diseases. We exploit the HODMD advantageous properties in dynamics identification and noise cleaning to identify the relevant frequencies and coherent patterns for each one of the diseases. The echocardiography datasets consist of video loops taken with respect to a long axis view (LAX) and a short axis view (SAX), where each video loop covers at least three cardiac cycles, formed by (at most) 300 frames each (called snapshots). The proposed algorithm, using only a maximum quantity of 200 snapshots, was able to capture two branches of frequencies, representing the heart rate and respiratory rate. Additionally, the algorithm provided a number of modes, which represent the dominant features and patterns in the different echocardiography images, also related to the heart and the lung. Six datasets were analyzed: one echocardiography taken from a healthy subject and five different sets of echocardiography taken from subjects with either Diabetic Cardiomyopathy, Obesity, SFSR4 Hypertrophy, TAC Hypertrophy or Myocardial Infarction. The results show that HODMD is robust and a suitable tool to identify characteristic patterns able to classify the different pathologies studied.

On the topology patterns and symmetry breaking in two planar synthetic jets

This article studies the flow structures and main patterns driving the flow dynamics in one and two planar synthetic jets. We perform numerical simulations at different Reynolds numbers (Re), for a similar forcing frequency, to model the synthetic jet flow and the two planar synthetic jets, which present a movement in-phase (synchronous jets) and out-of-phase (asynchronous jets). We identify two types of flow regimes as function of the Reynolds number: (i) the flow is symmetric and (ii) the symmetry is broken at Re ≈110 and Re ≈140 for the single and the synchronous jets, respectively. On the contrary, the flow is always asymmetric in the two asynchronous jets. We calculate the thrust produced by the several jet configurations, finding that the thrust produced by a single jet is always half of the thrust produced by the two synchronous jets; however, this quantity is much smaller in the asynchronous jets. Finally, we use higher order dynamic mode decomposition to identify the main patterns driving the flow dynamics. The solution is periodic in the single and two synchronous jets, with the forcing frequency (St*) as the dominant mode. The emerging rise in amplitude of a low-frequency mode (St0 = St*/6) that sub-harmonic of the forcing frequency as the Reynolds number increases suggests a connection between this mode and the symmetry breaking. A new mode is identified in the asynchronous jets, breaking the flow periodicity.

Application of higher order dynamic mode decomposition to modal analysis and prediction of power systems with renewable sources of energy

Concern for climate change is driving a vastly increased use of electricity and variable renewable energy supply encourages larger and evermore interconnected power systems. Stability analysis and short-term prediction of power system output has never been more urgent or more complex. Many distributed and renewable generators contribute zero inertia to the system and increase the risk of poorly damped oscillations leading to cascading outage. Data-driven techniques, higher order dynamic mode decomposition (HODMD) and total-least-squares higher-order dynamic mode decomposition (THDMD) are applied to modal analysis and short-term prediction of frequency and power exchange deviations. The decomposition uses multiple and randomized sampling windows of historical measurements. Dominant THDMD and HODMD modes can be used to show the contribution of renewable generation, such as wind power, to wide-area oscillations. The developed techniques are applied to the analysis of blackouts in Europe (2006) and the UK (2019), as well as the separation event in Australia (2018). The obtained results demonstrate that the damping of some HODMD modes can be overestimated. Although selected HODMD modes can reconstruct and predict power system output, the results are not always reliable. In turn, THDMD can predict dominant oscillations, with reduction of noise bias error in modal analysis of noisy measurements. With low noise data both techniques can produce very similar modal results.

Three-dimensional flow field reconstruction in the wake of a confined square cylinder using planar PIV data

A method is presented to perform the three-dimensional reconstruction of the velocity field in laminar flows using planar PIV data. The developed method relies on a convenient, repeated application of two data processing tools developed by two of the authors and called the higher order dynamic mode decomposition and the spatio-temporal Koopman decomposition. The present reconstruction method is an extension of a general method recently presented by three of the authors (Pérez et al., 2020), in which synthetic PIV data (obtained via numerical simulation) along three mutually orthogonal families of planes were used. In this article, instead, actual experimental data are considered, which requires additional data processing tools, developed in the article, to both cope with experimental noise and synchronize the various PIV datasets obtained in different timespans. The method is applied to analyze the wake of a square cylinder confined in a channel, at moderate Reynolds number. Due to limitations of the experimental setup used to obtain data for this configuration, only two families of planar PIV measurements are available that are contained in planes parallel to the sides of the channel. Because of this limitation, only the streamwise velocity component is three-dimensionally reconstructed. The performance of the method is tested in the wake of the confined square cylinder, comparing the reconstructed data with experimental PIV data measured on additional test planes (not used in the reconstruction). The obtained reconstructions of the streamwise velocity are reasonably accurate.

Uncovering spatiotemporal patterns in semiconductor superlattices by efficient data processing tools.

Time periodic patterns in a semiconductor superlattice, relevant to microwave generation, are obtained upon numerical integration of a known set of drift-diffusion equations. The associated spatiotemporal transport mechanisms are uncovered by applying (to the computed data) two recent data processing tools, known as the higher order dynamic mode decomposition and the spatiotemporal Koopman decomposition. Outcomes include a clear identification of the asymptotic self-sustained oscillations of the current density (isolated from the transient dynamics) and an accurate description of the electric field traveling pulse in terms of its dispersion diagram. In addition, a preliminary version of a data-driven reduced order model is constructed, which allows for extremely fast online simulations of the system response over a range of different configurations.

A new automatic, very efficient method for the analysis of flight flutter testing data

This article presents a new data processing method, based on higher order dynamic mode decomposition, to analyze actual flight test experimental data. The new methodology (i) is fast and efficient, (ii) is automatic, (iii) provides a clean spectrum in terms of the frequencies and damping rates of the flutter modes, and (iv) reconstructs the spatial distribution of the modes over the aircraft surface. The method gives useful and interesting information for predicting flutter and is suitable to analyze flight test data in real time.

Numerical analysis on modal stability characteristics of 2D panel flutter at low supersonic speeds

In order to better understand the mechanism of panel flutter at low supersonic speeds, and to study the viscous effect on flutter boundary and flutter type, a coupled computational fluid dynamics and computational structure dynamics (CFD/CSD) methodology has been adopted. Detailed modal stability analyses have been conducted through the higher-order dynamic mode decomposition (HODMD) method, which has been used to extract dynamic modes from CFD/CSD time-domain data. The results show that at low supersonic speeds, for relatively small dynamic pressure, the stability of the first mode increases due to viscous effect and decreases when the dynamic pressure exceeds a certain critical value. For other higher-order modes the flow viscosity has a stabilizing effect when the dynamic pressure is small, and has a destabilizing when the dynamic pressure exceeds a certain value. The mode stability variation causes the difference of flutter boundary and flutter type between viscous and inviscid solutions at low supersonic speeds. Moreover, it is found that for a simply supported 2D panel, the higher-order single-mode flutter can still occur in the turbulent flow around Mach 1.4 within certain structural parameter values, especially when the mass ratio is small. The high frequency of this type of flutter suggest that more severe structural fatigue problem may occur on simply supported panels with large aspect ratio at certain parameter values, thus requiring due attention.

Hydrodynamical study of terahertz emission in magnetized graphene field-effect transistors

Several hydrodynamic descriptions of charge transport in graphene have been presented in recent years. We discuss a general hydrodynamic model governing the dynamics of a two-dimensional electron gas in a magnetized field-effect transistor in the slow drift regime. The Dyakonov–Shur instability is investigated, including the effect of weak magnetic fields (i.e., away from Landau levels). We verify that the occurrence of the gap on the dispersion relation imposes a limit on the Mach number of the electronic flow, as it does not allow the unstable frequencies to be below ωc. Furthermore, we discuss that the presence of the external magnetic field decreases the growth rate of the instability, as well as the saturation amplitude. The numerical results from our simulations and the presented higher order dynamic mode decomposition support such reasoning.

Higher order dynamic mode decomposition of wind pressures on square buildings

Wind pressures on buildings with different aspect ratios were investigated via higher-order dynamic mode decomposition (HODMD) in this study. Taken’s embedding theorem was used to augment the spatial dimensionality of the original snapshot matrix by introducing time delay coordinates. HODMD was applied to reveal the spatial-temporal evolution characteristics of fluctuating wind pressures on building surfaces. It was found that HODMD can successfully extract the modes and corresponding frequencies from the random pressure field. As the aspect ratio increases, the main mode is more dominant than other modes. Moreover, comparisons between HODMD and proper orthogonal decomposition (POD) suggests that the first-order mode shapes extracted by HODMD and POD are generally similar, and both can reflect the main characteristics of the fluctuating pressure field. However, the higher-order mode shapes extracted by HODMD and POD are different due to higher-order nonlinearities. The modes of HODMD oscillate at a fixed frequency while those of POD resemble random signals, indicating the physical meaning of HODMD is more evident. Furthermore, HODMD is proven to be superior to POD in reconstructing pressure fields with less root mean square errors, because HODMD reconstructs the pressure field directly while POD primarily aims to rebuild the energy field.

Flow Structures on a Planar Food and Drug Administration (FDA) Nozzle at Low and Intermediate Reynolds Number

In this paper, we present a general description of the flow structures inside a two-dimensional Food and Drug Administration (FDA) nozzle. To this aim, we have performed numerical simulations using the numerical code Nek5000. The topology patters of the solution obtained, identify four different flow regimes when the flow is steady, where the symmetry of the flow breaks down. An additional case has been studied at higher Reynolds number, when the flow is unsteady, finding a vortex street distributed along the expansion pipe of the geometry. Linear stability analysis identifies the evolution of two steady and two unsteady modes. The results obtained have been connected with the changes in the topology of the flow. Finally, higher-order dynamic mode decomposition has been applied to identify the main flow structures in the unsteady flow inside the FDA nozzle. The highest-amplitude dynamic mode decomposition (DMD) modes identified by the method model the vortex street in the expansion of the geometry.

Origins of atypical shock buffet motions on a supercritical aerofoil

To expand the allowable flight regime of contemporary aircraft, a fundamental understanding of edge-of-the-envelope flow phenomena, such as transonic shock buffet, is necessary. In this study, Unsteady Reynolds-Averaged Navier-Stokes simulations are employed to probe the evolution of shock motions throughout the buffet envelope. At deep-buffet freestream conditions, atypical shock motions are shown to develop, in which an alternating strong/weak shock formation is evident with significant upstream pressure wave propagation. The transition between typical, single-frequency shock oscillations and these high-periodicity aerodynamic responses with increasing incidence is found to exhibit characteristics of a period-doubling route to chaos. Analysis of these flows through Higher-Order Dynamic Mode Decomposition reveals the exotic buffet dynamics emerge from the interplay between a dominant oscillatory buffet mode and its related even sub- and superharmonics, which appear with equivalent modal energies at high-incidence conditions. Correspondingly, the tendency for upstream pressure wave propagation is found to be correlated to a reversal in the suction surface leading edge phase gradient of the dominant oscillatory modes, which also develops with increasing angle of attack.

Near and far field laminar flow structures in an axisymmetric zero-net-mass-flux jet

Numerical simulations have been performed to study the near and far fields of an axisymmetric zero-net-mass flux (ZNMF) jet in the laminar-transitional regime, before the three-dimensional vortex breakdown that promotes transition to turbulence. The main frequencies driving the flow and their associated spatial modes have been calculated using the iterative, multidimensional higher order dynamic mode decomposition algorithm. The results show that the flow is periodic in the near field, but it is quasi-periodic in the far field, where two additional high frequency traveling modes appear. One of them is localized downstream a saddle point (a topology pattern characteristic of the suction phase defining the jet flow oscillations) and seems to be responsible for a shear layer instability. The second high frequency traveling mode, instead, extends further downstream and is associated with the overall wake of the ZNMF jet, which behaves in this sense as continuous jet. Evidence of non-linear coupling between the different spatio-temporal modes is presented in this work, which suggests the existence of a mechanism of energy transition between the periodic modes in the near field and the quasi-periodic modes in the far field previous to transition.