Enhanced Subspace Dynamic Mode Decomposition for Operational Modal Analysis of Aerospace Structures

Pulsed high-intensity focused ultrasound (pHIFU) can induce sparse de novo inertial cavitation without the introduction of exogenous contrast agents, promoting mild mechanical disruption in targeted tissue. Because the bubbles are small and rapidly dissolve after each HIFU pulse, mapping transient bubbles and obtaining real-time quantitative metrics correlated to tissue damage are challenging. Prior work introduced Bubble Doppler, an ultrafast power Doppler imaging method as a sensitive means to map cavitation bubbles. The main limitation of that method was its reliance on conventional wall filters used in Doppler imaging and optimized for imaging blood flow rather than transient scatterers. This study explores Bubble Doppler enhancement using dynamic mode decomposition (DMD) of a matrix created from a Doppler ensemble for mapping and extracting the characteristics of transient cavitation bubbles. DMD was first tested in silico with a numerical dataset mimicking the spatiotemporal characteristics of backscattered signal from tissue and bubbles. The performance of DMD filter was compared to other widely used Doppler wall filters - singular value decomposition (SVD) and infinite impulse response (IIR) highpass filter. DMD was then applied to an ex vivo tissue dataset where each HIFU pulse was immediately followed by a plane wave Doppler ensemble. In silico DMD outperformed SVD and IIR high pass filter and ex vivo provided physically interpretable images of the modes associated with bubbles and their corresponding temporal decay rates. These DMD modes can be trackable over the duration of pHIFU treatment using k-means clustering method, resulting in quantitative indicators of treatment progression.

A parallel and streaming Dynamic Mode Decomposition algorithm with finite precision error analysis for large data

Understanding sloshing modes in cylindrical tanks with internal components is essential for risk mitigation. Moreover, the complex geometries make theoretical analysis difficult. While dynamic mode decomposition (DMD) has been widely adopted in fluid dynamics, it is meaningful to study its application in sloshing. Unlike conventional flow modes, DMD may face unique challenges due to the discontinuous free surface dynamics. To address this issue, this study selected a container with cylindrical components as the research object. The DMD method was employed to extract sloshing modes from numerical simulations using the Volume of Fluid method. The DMD accuracy is validated through comparison with simulation results. The Froude number (Fr) is used to assess DMD's robustness across different sloshing regimes. Results show that DMD method can effectively capture the sloshing modes. In linear sloshing, the overall modes are governed by the liquid-wall impact. When excited at the first order natural frequency, a nonlinear wave collision pattern is found and generates acceptable errors. Under the second order natural frequency excitation, the errors are sharply increased. Additionally, the relationship between DMD and Fr is independent of the tank filling rate. When Fr < 0.198, the DMD calculation result is reliable. For 0.198 ≤ Fr ≤ 0.368, DMD maintains high reconstruction accuracy, but the prediction error increases by 134.6%. When Fr > 0.368, both prediction and reconstruction errors surge by 128.3% and 127.5%. This work proposes a novel and effective framework in tank's sloshing analysis using DMD. The results provide valuable guidance for tank design, especially with complex geometries and internal components.

The full numerical resolution of planetary flows, with the complex interdependence of mesoscale and sub-mesoscale dynamics that characterize such large scale circulation, is beyond reach with nowadays technology. When performing a numerical simulation of the ocean or the atmosphere, great care must be put in the choice of the parametrization of all those scales that are too small to be efficiently resolved.This work investigates the benefits of a stochastic decomposition of the Lagrangian trajectory into a smooth-in-time large scale velocity and a random fast-evolving uncorrelated part, ideally accounting for mesoscale and submesoscale processes. This approach, named Location Uncertainty (LU) [1], is built upon a stochastic version of the Reynolds Transport Theorem allowing us to cast the classical physical conservation laws into this scale-separated framework. This framework has been proven to be successful in several large-scale models for ocean dynamics [2,3,4,5].The derivation and implementation (within the community model NEMO, https://www.nemo-ocean.eu) of the hydrostatic primitive equations in this stochastic framework has been outlined in [6] and it is tested in this work with a novel data-driven approach based on dynamical mode decomposition [7]. The flow prediction in an idealized double-gyre configuration is shown to be improved by this stochastic contribution.[1], E. Mémin Fluid flow dynamics under location uncertainty,(2014), Geophysical & Astrophysical Fluid Dynamics, 108, 2, 119–146.[1] W. Bauer, P. Chandramouli, B. Chapron, L. Li, and E. Mémin. Deciphering therole of small-scale inhomogeneity on geophysical flow structuration: a stochastic approach. Journal of Physical Oceanography, 50(4):983-1003, 2020.[2] W. Bauer, P. Chandramouli, L. Li, and E. Mémin. Stochastic representation ofmesoscale eddy effects in coarse-resolution barotropic models. Ocean Modelling, 151:101646, 2020.[3] Rüdiger Brecht, Long Li, Werner Bauer and Etienne Mémin. Rotating ShallowWater Flow Under Location Uncertainty With a Structure-Preserving Discretization. Journal of Advances in Modeling Earth Systems, 13, 2021MS002492.[5] V. Resseguier, L. Li, G. Jouan, P. Dérian, E. Mémin, B. Chapron, (2021), New trends in ensemble forecast strategy: uncertainty quantification for coarse-grid computational fluid dynamics, Archives of Computational Methods in Engineering.[6] F.L. Tucciarone, E. Mémin, L. Li, (2022), Primitive Equations Under Location Uncertainty: Analytical Description and Model Development, Stochastic Transport in Upper Ocean Dynamics, Springer.[7] L. Li, E. Mémin, G. Tissot,  Stochastic Parameterization with Dynamic Mode Decomposition, Stochastic Transport in Upper Ocean Dynamics, Springer.

Dynamic mode decomposition (DMD) is essentially a hybrid algorithm based on mode decomposition and singular value decomposition, and it inevitably inherits the drawbacks of these two algorithms, including the selection strategy of truncated rank order and wanted mode components. A novel denoising and feature extraction algorithm for multi-component coupled noisy mechanical signals is proposed based on the standard DMD algorithm, which provides a new method solving the two intractable problems above. Firstly, a sparse optimization method of non-convex penalty function is adopted to determine the optimal dimensionality reduction space in the process of DMD, obtaining a series of optimal DMD modes. Then, multiscale permutation entropy calculation is performed to calculate the complexity of each DMD mode. Modes corresponding to the noise components are discarded by threshold technology, and we reconstruct the modes whose entropies are smaller than a threshold to recover the signal. By applying the algorithm to rolling bearing simulation signals and comparing with the result of wavelet transform, the effectiveness of the proposed method can be verified. Finally, the proposed method is applied to the experimental rolling bearing signals. Results demonstrated that the proposed approach has a good application prospect in noise reduction and fault feature extraction.

We establish the convergence of a class of numerical algorithms, known as dynamic mode decomposition (DMD), for computation of the eigenvalues and eigenfunctions of the infinite-dimensional Koopman operator. The algorithms act on data coming from observables on a state space, arranged in Hankel-type matrices. The proofs utilize the assumption that the underlying dynamical system is ergodic. This includes the classical measure-preserving systems, as well as systems whose attractors support a physical measure. Our approach relies on the observation that vector projections in DMD can be used to approximate the function projections by the virtue of Birkhoff's ergodic theorem. Using this fact, we show that applying DMD to Hankel data matrices in the limit of infinite-time observations yields the true Koopman eigenfunctions and eigenvalues. We also show that the singular value decomposition, which is the central part of most DMD algorithms, converges to the proper orthogonal decomposition of observables. We use this result to obtain a representation of the dynamics of systems with continuous spectrum based on the lifting of the coordinates to the space of observables. The numerical application of these methods is demonstrated using well-known dynamical systems and examples from computational fluid dynamics.

In fluid mechanics, modal decomposition, deeply intertwined with the concept of symmetry, is an essential data analysis method. It facilitates the segmentation of parameters such as flow, velocity, and pressure fields into distinct modes, each exhibiting symmetrical or asymmetrical characteristics in terms of amplitudes, frequencies, and phases. This technique, emphasizing the role of symmetry, is pivotal in both theoretical research and practical engineering applications. This paper delves into two dominant modal decomposition methods, infused with symmetry considerations: Proper Orthogonal Decomposition (POD) and Dynamic Mode Decomposition (DMD). POD excels in dissecting flow fields with clear periodic structures, often showcasing symmetrical patterns. It utilizes basis functions and time coefficients to delineate spatial modes and their evolution, highlighting symmetrical or asymmetrical transitions. In contrast, DMD effectively analyzes more complex, often asymmetrical structures like turbulent flows. By performing iterative analyses on the flow field, DMD discerns symmetrical or asymmetrical statistical structures, assembling modal functions and coefficients for decomposition. This method is adapted to extracting symmetrical patterns in vibration frequencies, growth rates, and intermodal coupling. The integration of modal decomposition with symmetry concepts in fluid mechanics enables the effective extraction of fluid flow features, such as symmetrically or asymmetrically arranged vortex configurations and trace evolutions. It enhances the post-processing analysis of numerical simulations and machine learning approaches in flow field simulations. In engineering, understanding the symmetrical aspects of complex flow dynamics is crucial. The dynamics assist in flow control, noise suppression, and optimization measures, thus improving the symmetry in system efficiency and energy consumption. Overall, modal decomposition methods, especially POD and DMD, provide significant insights into the symmetrical and asymmetrical analysis of fluid flow. These techniques underpin the study of fluid mechanics, offering crucial tools for fluid flow control, optimization, and the investigation of nonlinear phenomena and propagation modes in fluid dynamics, all through the lens of symmetry.

Lagrangian dynamic mode decomposition for construction of reduced-order models of advection-dominated phenomena

The modes in the electrocardiogram (ECG) signal can be divided into stable modes and unstable modes. The unstable modes are of great significance for signal analysis and classification. When the traditional dynamic mode decomposition (DMD) method is directly applied to these signals, it is difficult to effectively extract these key modes due to the rank mismatch problem of the data matrix. In order to better capture and analyze unstable modes, this study introduced the Hankel matrix to expand ECG data and used it as the input of DMD to propose the Hankel dynamic mode decomposition (HDMD) method. Although HDMD has been applied in other physiological signal processing, this study is the first to successfully apply it to multi-lead ECG signal analysis. By optimizing the delay parameters of the Hankel matrix and retaining the modes that account for 90% of the singular value decomposition energy, we significantly improve the effectiveness of feature extraction. Each type of abnormal ECG signal data is classified on the PTB Diagnostic ECG database. The experimental results of the proposed model are compared with the direct use of DMD for modal extraction. The highest classification accuracy of multi-modal ECG signal data extracted by HDMD is more than 10% higher than that using DMD. At the same time, the mean squared error (MSE) of the reconstruction using HDMD is 0.282 lower than that of DMD, indicating a significant improvement in reconstruction accuracy. Further illustrating the proposed method, the HDMD method can better capture and analyze the unstable modes in the ECG signal, thereby significantly improving the accuracy and robustness of signal classification. Future work will focus on further optimizing the parameter selection of the HDMD model and exploring its application potential in real-time heart disease monitoring and early warning systems.

The dynamic mode decomposition (DMD) method improved by embedded dimension and the proper orthogonal decomposition (POD) method are used to analyze the random wind pressure field of a large-span flat roof. The results show that the fuzzy dynamic features hidden in the wind tunnel test data can be discovered by using the Takens embedding theorem to increase the spatial dimension of the original snapshot matrix. The modal distributions of POD and DMD can reflect the main characteristics of typical vortices in a fluctuating pressure field. However, each mode of POD contains the information of multiple frequencies, which to a certain extent makes the mode of POD become the fluctuating coupling of multiple frequency bands. The mode decomposed by DMD method is a single-frequency mode and the stability of each mode can be obtained. At the same proportion, POD modes contain more energy than DMD modes. But when the same proportion of modes are used to reconstruct the fluctuating wind pressure field, the reconstruction results of DMD method can better describe and fit the local characteristics of the original fluctuating pressure field than those of POD method. This is because DMD method reconstructs the pressure field directly, while POD method mainly reconstructs the energy field. Therefore, the DMD method has more advantages in revealing the spatial-temporal evolution characteristics of a random wind pressure field.

The cavitation of the tip leakage vortex (TLV) induced by tip leakage has always been a difficult problem faced by turbomachinery, and its flow structure is complex and diverse. How to accurately extract the main structures that affect the cavitating flow of the TLV from the two-phase flow field is a key problem. In this study, the main mode extraction and low order mode reconstruction accuracy of the cavitation flow field of TLV downstream of National Advisory Committee for Aeronautics (NACA)0009 hydrofoil by two dynamic mode decomposition (DMD) methods are compared. The research shows that the main modes extracted by the standard DMD method contain a large number of noise modes, while the sparsity-promoting DMD eliminates the noise modes, showing obvious advantages in the reconstruction accuracy of the velocity field. The characteristics of cavitation signals are analyzed, and the cavitation signals are divided into four categories, which explains the reason why DMD methods have low reconstruction accuracy in cavitation. This study provides a theoretical basis and strong guarantee for the extraction of mode decomposition characteristics of the two-phase flow field. This is of great significance for accelerating the prediction of multiphase flow fields based on intelligent flow pattern learning in the future. Meanwhile, it also provides a new method and road for the introduction of artificial intelligence technology in future scientific research.

The objective of this paper is to identify the dominant coherent structures within cavitating flow around a Clark-Y hydrofoil using two data-driven modal decomposition methods, proper orthogonal decomposition (POD) and dynamic mode decomposition (DMD). A snapshot data sequence is obtained using a large eddy simulation and the interaction between cavitation and the vortex during cloud cavity shedding evolution is investigated. Modal decomposition via POD and DMD indicates that the dominant coherent structures include the large-scale cavity–vortex, re-entrant jet, shear layer, and small-scale vortex in the wake. In addition, the flow field can be reconstructed from the most energetic POD or DMD modes. The errors in the flow reconstructions produced using the first four POD modes, first eight POD modes, and first eight DMD modes are 3.884%, 3.240%, and 3.889%, respectively. Furthermore, transient cavitating flow can be predicted via the DMD method with an error of 8.081%. The largest errors in the reconstructed and predicted results occur mostly in the shear layer, trailing edge, and near wake. POD and DMD provide accurate and practically beneficial techniques for understanding cavitating flow, although substantial challenges remain with regard to predicting this intense nonlinear system.

Recently, acoustic communication employing orbital angular momentum (OAM) opens another avenue for efficient data transmission in aquatic environments. Current topological charge (TC) detection of OAM beams relies on the orthogonality among different-order OAM beams. However, such strategy requires measurements of the complete azimuthal acoustic pressure, which inevitably reduces the efficiency and increases the bit error rate (BER). To address these challenges, this study proposes a modified dynamic modal decomposition (DMD) method by partially sampling the acoustic field for precise TC detection. Numerical simulations confirm the accuracy of this approach in extracting single or multiple TCs magnitudes within a partially sampled acoustic field. We theoretically compare the performance of the modified DMD approach with conventional orthogonal decoding method. Simulation results indicate that our modified DMD scheme exhibits lower BER under the same noise interference and is more robust to the array misalignment. This research introduces an efficient demodulation solution for acoustic OAM communication, offering potential benefits for simplifying receiver array design and enhancing long-distance underwater data transmission.

A new dynamic mode decomposition (DMD) method is introduced for simultaneous system identification and denoising in conjunction with the adoption of an extended Kalman filter algorithm. The present paper explains the extended-Kalman-filter-based DMD (EKFDMD) algorithm which is an online algorithm for dataset for a small number of degree of freedom (DoF). It also illustrates that EKFDMD requires significant numerical resources for many-degree-of-freedom (many-DoF) problems and that the combination with truncated proper orthogonal decomposition (trPOD) helps us to apply the EKFDMD algorithm to many-DoF problems, though it prevents the algorithm from being fully online. The numerical experiments of a noisy dataset with a small number of DoFs illustrate that EKFDMD can estimate eigenvalues better than or as well as the existing algorithms, whereas EKFDMD can also denoise the original dataset online. In particular, EKFDMD performs better than existing algorithms for the case in which system noise is present. The EKFDMD with trPOD, which unfortunately is not fully online, can be successfully applied to many-DoF problems, including a fluid-problem example, and the results reveal the superior performance of system identification and denoising.

Dynamic mode decomposition (DMD) is an emerging methodology that has recently attracted computational scientists working on nonintrusive reduced order modeling. One of the major strengths that DMD possesses is having ground theoretical roots from the Koopman approximation theory. Indeed, DMD may be viewed as the data-driven realization of the famous Koopman operator. Nonetheless, the stable implementation of DMD incurs computing the singular value decomposition of the input data matrix. This, in turn, makes the process computationally demanding for high dimensional systems. In order to alleviate this burden, we develop a framework based on sketching methods, wherein a sketch of a matrix is simply another matrix which is significantly smaller, but still sufficiently approximates the original system. Such sketching or embedding is performed by applying random transformations, with certain properties, on the input matrix to yield a compressed version of the initial system. Hence, many of the expensive computations can be carried out on the smaller matrix, thereby accelerating the solution of the original problem. We conduct numerical experiments conducted using the spherical shallow water equations as a prototypical model in the context of geophysical flows. The performance of several sketching approaches is evaluated for capturing the range and co-range of the data matrix. The proposed sketching-based framework can accelerate various portions of the DMD algorithm, compared to classical methods that operate directly on the raw input data. This eventually leads to substantial computational gains that are vital for digital twinning of high dimensional systems.