Pointwise Error Bounds in POD Methods Without Difference Quotients

Proper orthogonal decomposition method for multiscale elliptic PDEs with random coefficients

The tangential impingement of turbulent circular jets on the cylindrical surface in an annular chamber is numerically investigated based on the SAS-SST model. The main purpose is to understand the role of the coherent structures on the temperature field inside the chamber and the heat transfer coefficient field on the hub surface. Based on the jet diameter and inlet velocity, the jet Reynolds numbers are given as 285000, 332000, and 392000, respectively. The accuracy of the numerical simulation is verified by the experimental data. The spectrum information of the velocity magnitude at the monitor point is obtained by the fast Fourier transform (FFT) method. An improvement of the proper orthogonal decomposition (POD) method is known as the extended POD (EPOD) method, which is mainly suitable for analyzing the correlation between the flow field and the synchronized scalar or vector fields, such as the correlation between the velocity and the heat transfer coefficient fields. In present study, both the POD method and the EPOD method are applied and compared. The POD method allows to visualize the main modes in terms of velocity, temperature, and heat transfer coefficient fluctuations. With the help of the EPOD method, the contribution of velocity fluctuations to temperature fluctuations in the chamber and heat transfer fluctuations on the hub surface are further analyzed. For Rej = 392000, the first two velocity POD modes and the velocity EPOD modes that contribute the most to the temperature fluctuations are very similar, reflecting the same large-scale coherent structures. The first two velocity EPOD modes that contribute the most to the heat transfer coefficient fluctuations have a high proportion of energy in the flow field, corresponding to the deflection motion of the jet.

To analyze the complex spatiotemporal evolution law of the high-speed jet flow field in a liquid-ring pump ejector, both the classic and spectral proper orthogonal decomposition (SPOD) methods were introduced to decompose the transient flow field based on the numerical large eddy simulation. The proper orthogonal decomposition (POD) reduced order model (ROM) of the complex flow in the ejector was established to analyze the transient characteristics and reconstruct the flow field. The spatial mode characteristics of density, pressure and vorticity field, and the frequency domain characteristics of mode coefficients were analyzed. The results show that the spatiotemporal decoupling analysis of a high-speed jet flow field can be accomplished by the POD and SPOD methods. A large-scale symmetric coherent structure is formed in the shear layer of the first two modes of vorticity, and a small-scale antisymmetric structure is formed at the trailing edge of the jet flow for the third and fourth modes. This reflects the evolutionary characteristics of the jet shear vortex. There is a significant spatial correlation between the density and pressure modes, reflecting that the evolution of the two structures has important correlation characteristics. The spectral analysis of the mode coefficients shows that the dominant frequencies of 1250 Hz and 18 kHz correspond to the self-excited oscillation frequency of shock waves and the vortex shedding frequency. The POD-ROM can reconstruct the high-speed flow field of the ejector with enough accuracy using the first 30-order modes, and the root mean square error and mean absolute percentage error for the velocity prediction are 0.48% and 0.24%, respectively.

We have analyzed the computational dynamics of a complex one-dimensional structural system consisting of a number of alternating stiff and soft subsystems. In particular, the method of Proper Orthogonal Decomposition (POD) in the frequency domain has been applied to analyze the (single-frequency) steady state dynamics in terms of spectral amplitudes and spatial shapes of proper orthogonal modes. It is shown that the dominant POD modes of such a multi-body system are sensitive to imperfections. The processing of the computational dynamics by the spectral-POD method leads to a novel computation of the transfer function of the system.

PurposeWind loading calculations are currently performed according to the ASCE 7 standard. Values in this standard were estimated from simplified models that do not necessarily take into account relevant flow characteristics. Thus, the standard does not have provisions to handle the majority of rooftop photovoltaic (PV) systems. Accurate solutions for this problem can be produced using a full-fledged three-dimensional computational fluid dynamics (CFD) analysis. Unfortunately, CFD requires enormous computation times, and its use would be unsuitable for this application which requires real-time solutions. To this end, a real-time response framework based on the proper orthogonal decomposition (POD) method is proposed.Design/methodology/approachA real-time response framework based on the POD method was used. This framework used beforehand and off-line CFD solutions from an extensive data set developed using a predefined design space. Solutions were organized to form the basis snapshots of a POD matrix. The interpolation network using a radial-basis function (RBF) was used to predict the solution from the POD method given a set of values of the design variables. The results presented assume varying design variables for wind speed and direction on typical PV roof installations.FindingsThe trained POD–RBF interpolation network was tested and validated by performing the fast-algebraic interpolation to obtain the pressure distribution on the PV system surface and they were compared to actual grid-converged fully turbulent 3D CFD solutions at the specified values of the design variables. The POD network was validated and proved that large-scale CFD problems can be parametrized and simplified by using this framework.Originality/valueThe solar power industry, engineering design firms and the society as a whole could realize significant savings with the availability of a real-time in situ wind-load calculator that can prove essential for plug-and-play installation of PV systems. Additionally, this technology allows for automated parametric design optimization to arrive at the best fit for a set of given operating conditions. All these tasks are currently prohibited because of the massive computational resources and time required to address large-scale CFD analysis problems, all made possible by a simple but robust technology that can yield massive savings for the solar industry.

This paper presents a proper orthogonal decomposition (POD) method that uses dynamic basis functions. The dynamic functions are of a prescribed form and do not explicitly depend on time but rather on parameters associated with flow unsteadiness. This POD method has been developed for modeling nonlinear flows with deforming meshes but can also be applied to fixed meshes. The method is illustrated for subsonic and transonic flows in channels with fixed and deforming meshes. This method properly captured flow nonlinearities and shock motion for cases in which the classical POD method failed.

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.

A proper orthogonal decomposition (POD) method is used to interpolate the flow around an airfoil for various Mach numbers and angles of attack in the transonic regime. POD uses a few numerical simulations, called snapshots, to create eigenfunctions. These eigenfunctions are combined using weighting coefficients to create a new solution for different values of the input parameters. Since POD methods are linear, their interpolation capabilities are quite limited when dealing with flow presenting nonlinearities, such as shocks. In order to improve their performance for cases involving shocks, a new method is proposed using variable fidelity. The main idea is to use POD to interpolate the difference between the CFD solution obtained on two different grids, a coarse one and a fine one. Then, for any new input parameter value, a coarse grid solution is computed using CFD and the POD interpolated difference is added to predict the fine grid solution. This allows some nonlinearities associated with the flow to be introduced. Results for various Mach numbers and angles of attack are compared to full CFD results. The variable fidelity-based POD method shows good improvement over the classical approach.

This study presents experimental investigations of entrainment events of a single sediment particle resting in a small pocket on a smooth bed in an open channel flow. The data were acquired with a tomographic particle tracking velocimetry system. The shake the box algorithm, a spatially high resolved Lagrangian tracking method, was applied to determine time-resolved and three-dimensional flow velocities in a volume during particle entrainment. The proper orthogonal decomposition (POD) method was applied to identify motions in the flow field carrying the most turbulent kinetic energy (TKE). Based on the POD method, the most energetic flow structures were linked with the occurring quadrant events at particle entrainment. It was shown that particle entrainment follows TKE peaks based on low-order POD modes caused by large sweeps. Two streamwise elongated counter rotating vortices with emerging sweeps in between were observed during particle entrainment. Here, most of the TKE exists in the first POD mode. This signature is suggested to be part of very large-scale coherent motions.

Dynamic response analysis of a double-disc rotor system with rolling bearings based on POD method

This paper discusses the application of a nonlinear analysis method based on the proper orthogonal decomposition (POD) method and Galerkin projection to the analysis and characterization of nonlinear modal interaction in stressed power systems. Using Galerkin projections onto bases of eigenfunctions obtained from the POD method, a new model is constructed that determines approximate solutions to the nonlinear power system behavior. The practical use of these methodologies is tested on a two area, 4-generators test system model. The validation is based on comparison with results from transient stability analysis using a commercially available transient stability program.

The blade angle has a great effect on hydraulic performance and internal flow field for axial-flow pumps. This research investigated the effect of the blade angle on hydraulic performance and tip leakage vortex (TLV) of a slanted axial-flow pump. The hydraulic performance and the TLV are compared with different setting angles. The dimensionless turbulence kinetic energy (TKE) is used to investigate the TLV. A novel variable fv is utilized to analyze the relation among the TLV, strain tensor and vorticity tensor. The proper orthogonal decomposition (POD) method is used to analyze TLV structure. The results show that with the increase of the blade angle, the pump head is getting larger, the flow rate of the best efficiency moves to be larger, and both the primary TLV (P-TLV) and the secondary TLV (S-TLV) are getting stronger. The P-TLV often exists in the outer edge of TKE distribution and S-TLVs often exist in the largest value area of TKE. This phenomenon is more evident with blade angle increasing. Through POD method, it shows that the first six modes contain more than 90% of TKE. The reason why the TKE value near the region of S-TLV is high is that the tip leakage flow is a kind of jet-like flow with high kinetic energy. The main structure of the P-TLV is shown in modes 4−6, resulting in a reflux zone but not with the highest TKE.

In a recent work (Koc et al., SIAM J. Numer. Anal. 59(4), 2163–2196, 2021), the authors showed that including difference quotients (DQs) is necessary in order to prove optimal pointwise in time error bounds for proper orthogonal decomposition (POD) reduced order models of the heat equation. In this work, we introduce a new approach to including DQs in the POD procedure. Instead of computing the POD modes using all of the snapshot data and DQs, we only use the first snapshot along with all of the DQs and special POD weights. We show that this approach retains all of the numerical analysis benefits of the standard POD DQ approach, while using a POD data set that has approximately half the number of snapshots as the standard POD DQ approach, i.e., the new approach requires less computational effort. We illustrate our theoretical results with numerical experiments.

The application of the proper orthogonal decomposition (POD) method to the vibration response of a cracked rotor system is investigated. The covariance matrices of the horizontal and vertical whirl amplitudes are formulated based on the numerical and experimental whirl response data for the considered cracked rotor system. Accordingly, the POD is directly applied to the obtained covariance matrices where the proper orthogonal values (POVs), and the proper orthogonal modes (POMs) are obtained for various crack depths, unbalance force vector angles, and rotational speeds. It is observed that both POVs and their corresponding POMs are highly sensitive to the appearance of the crack and the unbalance force angle direction in the neighborhoods of the critical rotational speeds. The sensitivity zones of the POVs and POMs to the crack propagation are found to be coinciding with the unstable zones found by the Floquet's theory of the considered cracked system.

Due to the high specific stiffness and strength, composite shells have been widely used in fuel tanks of launch vehicles. The buckling analysis of composite shells with cutouts based on the finite element (FE) method is too time-consuming. From the point-of-view of model size reduction, a novel Proper Orthogonal Decomposition (POD)-based buckling method is proposed in this paper, which can significantly increase the computational efficiency of buckling analysis. In order to improve the efficiency and effectiveness of prediction and optimization of composite shells with multiple cutouts, the POD method is integrated into an optimization framework that uses Gaussian process (GP) machine learning method. First, the training set used for the machine learning training is generated efficiently by means of the POD method. Then, the obtained set is trained and tested based on the Gaussian process method. The inputs are ply angles of the composite shell and the output is the buckling load of the composite shell containing cutouts. In order to maximize the buckling load of the composite shell against cutouts, the Genetic Algorithm is combined with the trained Gaussian process method to search for the optimal ply angles. Finally, an illustrative example is carried out to demonstrate the effectiveness of the proposed prediction and optimization framework.