Proper Orthogonal Decomposition Model Research Articles

Three-Dimensional Nature of Low-Frequency Unsteadiness in a Turbulent Separation Bubble

Three-dimensional behavior of low-frequency unsteadiness in the incompressible turbulent separation bubble (TSB) produced by a wall-mounted hump is investigated using time-resolved planar and stereoscopic particle image velocimetry measurements at several planes across the separated region. The aspect ratio (wind tunnel width/separation length, Lz,WT/Lb=4.4) provides nominally two-dimensional flow for more than half of the spanwise extent of the test section. Analysis in the streamwise/wall-normal plane along the center of the test section shows low-frequency (St<0.1) large-scale motion of the separated region. The flowfield contains features of both geometry-induced and pressure-gradient-induced separation, but unsteady dynamics produce dominant frequencies closer to geometry-induced TSBs (St≈0.1) compared to purely pressure-gradient-induced TSBs (St≈0.01). Measurements along the spanwise direction and parallel to the initial shear layer development show strong evidence that the low-frequency motion is inherently three-dimensional, providing an additional dimension to the understanding of the flapping/breathing typically observed in planar streamwise/wall-normal measurements. Spectral proper orthogonal decomposition and low-order modeling identify spanwise undulations with wavelengths of the order of Lb and frequencies of St<0.1. The three-dimensional behavior causes a peak/valley formation along the span and a more localized expansion/contraction, leading to only small variations in the integral volume of the TSB.

Order reduction of special valve models based on pod method

Abstract The opening size of the disc-type special regulating valve (referred to as the special valve) can be used to regulate the fluid pressure and flow inside the cabin, and is an important equipment in the intake system of the high-altitude simulation test bench (referred to as the high-altitude bench). Due to the high dimensionality and complex solution of the special valve model, the calculation time is relatively long. In this paper, the Proper Orthogonal Decomposition (POD) model reduction method is applied to the mathematical model of the special valve, and the POD basis matrix is constructed through snapshots, thus obtaining a low-dimensional model and improving computational efficiency. The research results indicate that, while ensuring sufficient accuracy, this method can reduce the dimensionality of the special valve model, improve the computational speed of the simulator in calculating this case, and verify the effectiveness of the POD method.

A Multi-Fidelity Uncertainty Propagation Model for Multi-Dimensional Correlated Flow Field Responses

Given the randomness inherent in fluid dynamics problems and limitations in human cognition, Computational Fluid Dynamics (CFD) modeling and simulation are afflicted with non-negligible uncertainties, casting doubts on the credibility of CFD. Scientifically and rigorously quantifying the uncertainty of CFD is paramount for assessing its credibility and informing engineering decisions. In order to quantify the uncertainty of multidimensional flow field responses stemming from uncertain model parameters, this paper proposes a method based on Gappy Proper Orthogonal Decomposition (POD) for supplementing high-fidelity flow field data within a framework that leverages POD and surrogate models. This approach enables the generation of corresponding high-fidelity flow fields from low-fidelity ones, significantly reducing the cost of high-fidelity flow field computation in uncertainty propagation modeling. Through an analysis of the impact of uncertainty in the coefficients of the Spalart–Allmaras (SA) turbulence model on the distribution of wall friction coefficients for the NACA0012 airfoil and pressure coefficients for the M6 wing, the proposed multi-fidelity modeling approach is demonstrated to offer significant advancements in both accuracy and efficiency compared to single-fidelity methods, providing a robust and efficient prediction model for large-scale random sampling.

Model order reduction for the 1D Boltzmann-BGK equation: identifying intrinsic variables using neural networks

Kinetic equations are crucial for modeling non-equilibrium phenomena, but their computational complexity is a challenge. This paper presents a data-driven approach using reduced order models (ROM) to efficiently model non-equilibrium flows in kinetic equations by comparing two ROM approaches: proper orthogonal decomposition (POD) and autoencoder neural networks (AE). While AE initially demonstrate higher accuracy, POD’s precision improves as more modes are considered. Notably, our work recognizes that the classical POD model order reduction approach, although capable of accurately representing the non-linear solution manifold of the kinetic equation, may not provide a parsimonious model of the data due to the inherently non-linear nature of the data manifold. We demonstrate how AEs are used in finding the intrinsic dimension of a system and to allow correlating the intrinsic quantities with macroscopic quantities that have a physical interpretation.

Thermal field reconstruction and compressive sensing using proper orthogonal decomposition

Model order reduction allows critical information about sensor placement and experiment design to be distilled from raw fluid mechanics simulation data. In many cases, sensed information in conjunction with reduced order models can also be used to regenerate full field variables. In this paper, a proper orthogonal decomposition (POD) inferencing method is extended to the modeling and compressive sensing of temperature, a scalar field variable. The method is applied to a simulated, critically stable, incompressible flow over a heated cylinder (Re = 1000) with Prandtl number varying between 0.001 and 50. The model is trained on pressure and temperature data from simulations. Field reconstructions are then generated using data from selected sensors and the POD model. Finally, the reconstruction error is evaluated across all Prandtl numbers for different numbers of retained modes and sensors. The predicted trend of increasing reconstruction accuracy with decreasing Prandtl number is confirmed and a Prandtl number/sensor count error matrix is presented.

A discontinuous Galerkin Method based on POD model reduction for Euler equation

&lt;abstract&gt; &lt;p&gt;This paper considers the work of combining the proper orthogonal decomposition (POD) reduced-order method with the discontinuous Galerkin (DG) method to solve three-dimensional time-domain Euler equations. The POD-DG formulation is established by constructing the POD base vector space, based on POD technology one can apply the Galerkin projection of the DG scheme to this dimension reduction space for calculation. Its overall goal is to overcome the disadvantages of high computational cost and memory requirement in the DG algorithm, reduce the degrees of freedom (DOFs) of the calculation model, and save the calculation time while ensuring acceptable accuracy. Numerical experiments verify these advantages of the proposed POD-DG method.&lt;/p&gt; &lt;/abstract&gt;

Flow sensing method for fluid-structure interaction systems via multilayer proper orthogonal decomposition

Flow sensing is widely used to forecast flow field in fluid-structure interaction (FSI) systems. The FSI system with multiple flexible structures usually involves complex unsteady flow and large number of sensors, which makes it difficult to perform flow sensing. In this study, a flow sensing method for FSI systems via multilayer proper orthogonal decomposition (POD) is proposed to achieve real-time forecast of flow field using structural deformation. First, we establish the POD model of structural deformation. To improve model accuracy for flow field, we propose the multilayer POD model, which mainly focus on the local modeling accuracy in the region with complex flow structures. Then, we establish the multilayer model of flow field. Furthermore, the deep neural network model is employed to map the mode coefficients of the structure to all the mode coefficients of the multilayer POD. The proposed method is applied in two FSI systems, including the flow past a flexible filament clamped behind a cylinder and the flow past flexible filament set. Both constant inflow and transient flow conditions are considered. The results indicate that the proposed flow sensing method exhibits excellent spatial-temporal performance, performs accurately in flow properties forecasting, and is suitable for FSI systems with complex flow structures such as coherent vortices.

An Optimal Control Method for Greenhouse Climate Management Considering Crop Growth’s Spatial Distribution and Energy Consumption

The environmental factors of greenhouses affect crop growth greatly and are mutually coupled and spatially distributed. Due to the complexity of greenhouse climate modeling, the current optimal control of greenhouse crop growth rarely considers the spatial distribution issues of environmental parameters. Proper Orthogonal Decomposition (POD) is a technique to reduce the order of a model by projecting it onto an orthogonal basis. In this paper, POD is used to extract environmental features from Computational Fluid Dynamics (CFD) simulations, and a low-dimensional feature subspace is obtained by energy truncation. With multi-dimensional interpolation, fast and low-dimensional reconstruction of the dynamic variation of greenhouse climates is achieved. On this basis, a rolling-horizon optimal control scheme is proposed. For each finite horizon, the external meteorological data are updated, and the response of the greenhouse environment is quickly calculated by the POD model. With the performance criterion J of maximizing crop production and energy efficiency, through the particle swarm optimization algorithm, the optimal settings for the greenhouse shading rate and the fan speed are derived. Such control computations are rolled forward during the whole planting season. Results of a case study show that the proposed method has low computation cost and high spacial resolution and can effectively improve the spatiotemporal accuracy of greenhouse climate management. In addition, different from traditional global optimal control methods, the proposed rolling-horizon scheme can correct various external disturbances in the procedure of crop growth, and thus it is more robust and has potential for engineering applications.

Airfoil-based convolutional autoencoder and long short-term memory neural network for predicting coherent structures evolution around an airfoil

Airfoil stall induced by unsteady flow separation is a major inhibitor that constrains the aircraft design. Hence, it is critical to have a better understanding of the flow dynamics around the airfoil near stall conditions. Recently, machine learning (ML) has been widely used to predict the spatiotemporal evolutions of coherent structures over the airfoil. Nonetheless, due to the existence of strong nonlinearity introduced by turbulence, most of the existing approaches focus on prediction of the flow fields in the laminar flow regime. In this work, we seek to examine the applicability of deep neural network (DNN) to prediction of turbulent flows around a NACA (National Advisory Committee for Aeronautics) 4412 airfoil. Large eddy simulations (LES) are performed to prepare the training dataset. Both proper orthogonal decomposition (POD) and DNN-based reduced-order models (ROM) are employed for reconstructing the flow fields. A convolutional autoencoder (CAE) is applied to map the high-dimensional flow dynamics into the low-dimensional nonlinear manifold. The geometric features are considered in the input matrix of the CAE network by setting physical quantities inside the airfoil to zero to reflect the physics. Then, the multi-head attention-strengthened long short-term memory (MH-LSTM) is utilized to predict the evolutions of latent vectors. Finally, the high-dimensional flow dynamics can be reconstructed by utilizing the decoder part of CAE. We believe that the MH-LSTM methodology provides a promising path towards predicting the unsteady flow field over the airfoil by capitalizing on the capability of multi-head attention allowing for attending to distinct parts of input sequence.

Analysis of Surrogate Models for Vapour Transport and Distribution in a Hollow Fibre Membrane Humidifier

To achieve high efficiency and low degradation of a polymer electrolyte fuel cell (PEMFC), it is necessary to maintain an appropriate level of humidification in the fuel cell membrane. Thus, membrane humidifiers are typically used in PEMFC systems. Parameter studies are important to evaluate membrane humidifiers under various operating conditions to reduce the amount of physical tests. However, simulative studies are computationally expensive when using detailed models. To reduce the computational cost, surrogate models are set up. In our study, a 3D computational fluid dynamics (CFD) model of a hollow fibre membrane humidifier is presented and validated using measurement data. Based on the results of the validated CFD model, a surrogate model of the humidifier is constructed using proper orthogonal decomposition (POD) in combination with different interpolation methods. To evaluate the surrogate models, their results are compared against reference solutions from the CFD model. Our results show that a Halton design combined with a thin-plate-spline interpolation results in the most accurate surrogate humidifier model. Its normalised mean absolute error for 18 test points when predicting the water mass fraction in the membrane humidifier is 0.58%. Furthermore, it is demonstrated that the solutions of the POD model can be used to initialise CFD calculations and thus accelerate the calculation of steady state CFD solutions.

A constrained proper orthogonal decomposition model for upscaling permeability

AbstractReservoir modeling and simulation are vital components of modern reservoir management processes. Despite the advances in computing power and the advent of smart technologies, the implementation of model‐based operational/control strategies has been limited by the inherent complexity of reservoir models. Thus, reduce order models that not only reduce the cost of the implementation but also provide geological consistent prediction are essential. This article introduces reduced‐order models based on the proper orthogonal decomposition (POD) coupled with linear interpolation for upscaling. First, using POD‐based models, low rank approximate (LRA) are obtained by projecting the high dimensional permeability dataset to a low dimensional subspace spanned by its trajectories to decorrelate the dataset. Next, the LRA is integrated into the interpolation algorithm to generate upscaled values. This technique is highly scalable since low‐rank approximations are achieved by the variation in the number of modes used for reconstruction. To test the validity and reliability of the model, we show its application to the practical dataset from SPE10 benchmark case2. From statistics of the error analysis, the classical POD algorithm seems to be more preferred for LRA; however, since non‐negativity of the permeability data set is a priority, the constrained POD (non‐negative POD) algorithm described in this article is more appropriate. Finally, we compared the POD‐based models to a traditional industry‐standard upscaling technique (e.g., arithmetic mean) to highlight our model benefits/performance. Results show that the POD‐based models, particularly the non‐negative POD model, produce considerably less error than the arithmetic mean model in the upscaling process.

A Simplified Newton Method to Generate Snapshots for POD Models of Semilinear Optimal Control Problems

In PDE-constrained optimization, proper orthogonal decomposition (POD) provides a surrogate model of a (potentially expensive) PDE discretization, on which optimization iterations are executed. Because POD models usually provide good approximation quality only locally, they have to be updated during optimization. Updating the POD model is usually expensive, however, and therefore often impossible in a model-predictive control (MPC) context. Thus, reduced models of mediocre quality might be accepted. We take the view of a simplified Newton method for solving semilinear evolution equations to derive an algorithm that can serve as an offline phase to produce a POD model. Approaches that build the POD model with impulse response snapshots can be regarded as the first Newton step in this context. In particular, POD models that are based on impulse response snapshots are extended by adding a second simplified Newton step. This procedure improves the approximation quality of the POD model significantly by introducing a moderate amount of extra computational costs during optimization or the MPC loop. We illustrate our findings with an example satisfying our assumptions.

Trajectory-optimized cluster-based network model for the sphere wake

We propose a novel trajectory-optimized cluster-based network model (tCNM) for nonlinear model order reduction from time-resolved data following Li et al. [“Cluster-based network model,” J. Fluid Mech. 906, A21 (2021)] and improving the accuracy for a given number of centroids. The starting point is k-means++ clustering, which minimizes the representation error of the snapshots by their closest centroids. The dynamics is presented by “flights” between the centroids. The proposed trajectory-optimized clustering aims to reduce the kinematic representation error further by shifting the centroids closer to the snapshot trajectory and refining state propagation with trajectory support points. Thus, curved trajectories are better resolved. The resulting tCNM is demonstrated for the sphere wake for three flow regimes, including the periodic, quasi-periodic, and chaotic dynamics. The representation error of tCNM is five times smaller as compared to the approximation by the closest centroid. Thus, the error is at the same level as proper orthogonal decomposition (POD) of same order. Yet, tCNM has distinct advantages over POD modeling: it is human interpretable by representing dynamics by a handful of coherent structures and their transitions; it shows robust dynamics by design, i.e., stable long-time behavior; and its development is fully automatable, i.e., it does not require tunable auxiliary closure and other models.

POD-Kalman filtering for improving noninvasive 3D temperature monitoring in MR-guided hyperthermia.

BackgroundDuring resonance frequency (RF) hyperthermia treatment, the temperature of the tumor tissue is elevated to the range of 39–44°C. Accurate temperature monitoring is essential to guide treatments and ensure precise heat delivery and treatment quality. Magnetic resonance (MR) thermometry is currently the only clinical method to measure temperature noninvasively in a volume during treatment. However, several studies have shown that this approach is not always sufficiently accurate for thermal dosimetry in areas with motion, such as the pelvic region. Model‐based temperature estimation is a promising approach to correct and supplement 3D online temperature estimation in regions where MR thermometry is unreliable or cannot be measured. However, complete 3D temperature modeling of the pelvic region is too complex for online usage.PurposeThis study aimed to evaluate the use of proper orthogonal decomposition (POD) model reduction combined with Kalman filtering to improve temperature estimation using MR thermometry. Furthermore, we assessed the benefit of this method using data from hyperthermia treatment where there were limited and unreliable MR thermometry measurements.MethodsThe performance of POD–Kalman filtering was evaluated in several heating experiments and for data from patients treated for locally advanced cervical cancer. For each method, we evaluated the mean absolute error (MAE) concerning the temperature measurements acquired by the thermal probes, and we assessed the reproducibility and consistency using the standard deviation of error (SDE). Furthermore, three patient groups were defined according to susceptibility artifacts caused by the level of intestinal gas motion to assess if the POD–Kalman filtering could compensate for missing and unreliable MR thermometry measurements.ResultsFirst, we showed that this method is beneficial and reproducible in phantom experiments. Second, we demonstrated that the combined method improved the match between temperature prediction and temperature acquired by intraluminal thermometry for patients treated for locally advanced cervical cancer. Considering all patients, the POD–Kalman filter improved MAE by 43% (filtered MR thermometry = 1.29°C, POD–Kalman filtered temperature = 0.74°C). Moreover, the SDE was improved by 47% (filtered MR thermometry = 1.16°C, POD–Kalman filtered temperature = 0.61°C). Specifically, the POD–Kalman filter reduced the MAE by approximately 60% in patients whose MR thermometry was unreliable because of the great amount of susceptibilities caused by the high level of intestinal gas motion.ConclusionsWe showed that the POD–Kalman filter significantly improved the accuracy of temperature monitoring compared to MR thermometry in heating experiments and hyperthermia treatments. The results demonstrated that POD–Kalman filtering can improve thermal dosimetry during RF hyperthermia treatment, especially when MR thermometry is inaccurate.

Disciplinary proper orthogonal decomposition and interpolation for the resolution of parameterized multidisciplinary analysis

AbstractThis article proposes a new method for the resolution of parameterized coupled systems of equations, such as typical in multidisciplinary analysis (MDA). It is based on the use of disciplinary surrogate models within a multi‐query context. The main idea is to replace the costly disciplinary solvers of the MDA by Proper Orthogonal Decomposition and Interpolation models. The main challenge we address is the high‐dimensional coupling variables whose ranges are unknown. To overcome this issue, a training strategy is developed by uncoupling the disciplinary solvers from the MDA context. This new surrogate MDA called Disciplinary Proper Orthogonal Decomposition and Interpolation is iteratively enriched to solve the analysis with an estimation of the error made by the surrogate models. The disciplinary surrogate model which has the most influence on this error is determined by a sensitivity analysis and thus enriched. This approach allows to uncouple the disciplinary solvers during the training and enrichment phases. The approach is applied to various aeroelastic problems of an aircraft wing and allows to reduce by a factor of 5 the mean number of disciplinary solver calls needed for the resolution of the MDA.

Fast flow simulation method based on POD and BPNN

It is computationally expensive to perform a large number of flow analysis by using high-fidelity CFD simulation. The paper proposes a fast flow-analysis method based on the proper orthogonal decomposition(POD) and back propagation based neural network(BPNN). First samples are generated in the geometric parameter space. Then a POD model is built to map the high-dimensional flow-field data to low-dimensional base modal coefficients, and further a BPNN model is fitted from geometric parameters to base modal coefficients to achieve fast flow prediction. During constructing the POD and BPNN model, the partitioning strategy and K-means clustering are implemented to improve modeling efficiency, as well as to reduce model training time. The results of predicting the steady flow of variable geometries show that: at subsonic, the trained model possesses good accuracy, especially in predicting the pressure isolines of the flow field and the pressure coefficient distribution of the airfoil. The average prediction errors of the lift and drag coefficient are smaller than 0.4%, while they are smaller than 1.4% at transonic. The shock location can be well predicted as well.

On closures for reduced order models—A spectrum of first-principle to machine-learned avenues

For over a century, reduced order models (ROMs) have been a fundamental discipline of theoretical fluid mechanics. Early examples include Galerkin models inspired by the Orr–Sommerfeld stability equation and numerous vortex models, of which the von Kármán vortex street is one of the most prominent. Subsequent ROMs typically relied on first principles, like mathematical Galerkin models, weakly nonlinear stability theory, and two- and three-dimensional vortex models. Aubry et al. [J. Fluid Mech. 192, 115–173 (1988)] pioneered the data-driven proper orthogonal decomposition (POD) modeling. In early POD modeling, available data were used to build an optimal basis, which was then utilized in a classical Galerkin procedure to construct the ROM, but data have made a profound impact on ROMs beyond the Galerkin expansion. In this paper, we take a modest step and illustrate the impact of data-driven modeling on one significant ROM area. Specifically, we focus on ROM closures, which are correction terms that are added to the classical ROMs in order to model the effect of the discarded ROM modes in under-resolved simulations. Through simple examples, we illustrate the main modeling principles used to construct the classical ROMs, motivate and introduce modern ROM closures, and show how data-driven modeling, artificial intelligence, and machine learning have changed the standard ROM methodology over the last two decades. Finally, we outline our vision on how the state-of-the-art data-driven modeling can continue to reshape the field of reduced order modeling.

A non-intrusive space-time interpolation from compact Stiefel manifolds of parametrized rigid-viscoplastic FEM problems

This work aims to interpolate parametrized reduced order model (ROM) basis constructed via the proper orthogonal decomposition (POD) to derive a robust ROM of the system’s dynamics for an unseen target parameter value. A novel non-intrusive space-time (ST) POD basis interpolation scheme is proposed, for which we define ROM spatial and temporal basis curves on compact Stiefel manifolds. An interpolation is finally defined on a mixed part encoded in a square matrix directly deduced using the spacial part, the singular values and the temporal part, to obtain an interpolated snapshot matrix, keeping track of accurate space and temporal eigenvectors. Moreover, in order to establish a well-defined curve on the compact Stiefel manifold, we introduce a new procedure, the so-called oriented SVD. Such an oriented SVD produces unique right and left eigenvectors for generic matrices, for which all singular values are distinct. It is important to notice that the ST POD basis interpolation does not require the construction and the subsequent solution of a reduced-order FEM model as classically is done. Hence it is avoiding the bottleneck of standard POD interpolation which is associated with the evaluation of the nonlinear terms of the Galerkin projection on the governing equations. As a proof of concept, the proposed method is demonstrated with the adaptation of rigid-thermoviscoplastic finite element ROMs applied to a typical nonlinear open forging metal forming process. Strong correlations of the ST POD models with respect to their associated high-fidelity FEM counterpart simulations are reported, highlighting its potential use for near real-time parametric simulations using off-line computed ROM POD databases.

Study on characteristics of flow structure and generation of narrow-band in low-frequency noise for a slat

Low-frequency narrow-band noise is one of the important components in the slat noise. Based on the transient flow field analysis for multi-element airfoil 30P-30N, the main mode characteristics of the slat pressure fluctuation are analyzed by means of proper orthogonal decomposition (POD) and dynamic model decomposition (DMD), and the key feature modes and the distribution of narrow-band frequencies for the slat are obtained. Empirical mode decomposition (EMD) and cross-correlation analysis are used to reconstruct the signal by preserving the low frequency characteristics of the original transient pressure signal. The locations of noise sources of the low-frequency narrow-band are determined according to the delay time and geometric relationship for characteristic points. The results show that the low-frequency narrow-band sound sources are mainly concentrated on regions of feature modes with large pressure fluctuations, such as slat trailing edge, downstream of the shear layer and the recirculation zone. Meanwhile, there are still some sound sources of low-frequency narrow-band distributed in regions of non-featured modes. The key characteristics of these regions are the interaction of flow structure between the trailing edge of the slat and the leading edge of the main wing due to collision of the shear layer.

A greedy non-intrusive reduced order model for shallow water equations

In this work, we develop Non-Intrusive Reduced Order Models (NIROMs) that combine Proper Orthogonal Decomposition (POD) with a Radial Basis Function (RBF) interpolation method to construct efficient reduced order models for time-dependent problems arising in large scale environmental flow applications. The performance of the POD-RBF NIROM is compared with a traditional nonlinear POD (NPOD) model by evaluating the accuracy and robustness for test problems representative of riverine flows. Different greedy algorithms are studied in order to determine a near-optimal distribution of interpolation points for the RBF approximation. A new power-scaled residual greedy (psr-greedy) algorithm is proposed to address some of the primary drawbacks of the existing greedy approaches. The relative performances of these greedy algorithms are studied with numerical experiments using realistic two-dimensional (2D) shallow water flow applications involving coastal and riverine dynamics.