Proper Orthogonal Decomposition Approach Research Articles

POD approach for unsteady aerodynamic model updating

A method for aerodynamic model updating is proposed in this paper. The approach is based upon a correction of the eigenvalues of the reduced-order unsteady aerodynamic matrix through an optimization with objective function defined through the difference in the generalized aerodynamic forces or on the aeroelastic poles. The high-fidelity model in reduced-order form is obtained by the proper orthogonal decomposition (POD) technique applied to the computational fluid dynamics Euler-based formulation. Many of the methods that have been developed in the past years for simpler aeroelastic models that use, for example, doublet-lattice method aerodynamics, can be adopted for this purpose as well. However, this model is not able to capture shocks and flow separation in transonic flow. The proposed approach performs the updating of the aerodynamic model by imposing the minimization of a global error between target aerodynamic performances, namely experimental performances, and an aerodynamic model in reduced-order form via POD approach. After a general presentation of the application of the POD method to the linearized Euler equations, the optimization strategy is presented. First, a simple test on a 2D wing section with theoretical biased data is performed, and then, the performances of different optimization strategies are tested on a 3D model updated by wind tunnel data.

Developing a Reduced-Order Model of Nonsynchronous Vibration in Turbomachinery Using Proper-Orthogonal Decomposition Methods

The paper develops a reduced-order model of nonsynchronous vibration (NSV) using proper orthogonal decomposition (POD) methods. The approach was successfully developed and implemented, requiring between two and six POD modes to accurately predict computational fluid dynamics (CFD) solutions that are experiencing NSV. This POD method was first developed and demonstrated for a transversely moving, two-dimensional cylinder in cross-flow. Later, the method was used for the prediction of CFD solutions for a two-dimensional compressor blade. This research is the first to offer a POD approach to the reduced-order modeling of NSV in turbomachinery. Modeling NSV is especially challenging because NSV is caused by complicated, unsteady flow dynamics; this initial study helps researchers understand the causes of NSV, and aids in the future development of predictive tools for aeromechanical design engineers.

Thermal Modeling of Multi-Fin Field Effect Transistor Structure Using Proper Orthogonal Decomposition

An approach is proposed to project thermal behavior in a semiconductor integrated-circuit structure onto a functional space based on the proper orthogonal decomposition (POD). The approach substantially reduces the numerical degrees of freedom (DOF) needed for thermal simulations and requires no assumptions about physical geometry, dimensions, or heat flow paths. The POD approach is applied to a multi-fin FinFET structure having heat sources driven by power pulse excitations with time shifts, width variations, and amplitude modulations. The POD models were compared with detailed numerical simulations (DNS) and it was shown that the POD approach provides thermal solutions that were as accurate and detailed as the DNS. It offers a reduction in numerical DOFs by nearly six orders of magnitude to capture the peak temperatures in multi-fin FinFETs.

An implementation of independent component analysis for 3D statistical shape analysis

An implementation of the independent component analysis (ICA) technique for three-dimensional (3D) statistical shape analysis is presented. The capabilities of the ICA approach to uncover inherent shape features are first demonstrated through analysis of sets of artificially generated surfaces, and the nature of these features is compared to a more traditional proper orthogonal decomposition (POD) technique. For the surfaces generated, the ICA approach is shown to consistently extract surface features that closely resembled the original basis surfaces used to generate the artificial dataset, while the POD approach produces features that clearly mix the original basis. The details of an implementation of the ICA approach within a statistical shape analysis framework are then presented. Results are shown for the ICA decomposition of a collection of clinically obtained human right ventricle endocardial surfaces (RVES) segmented from cardiac computed tomography imaging, and these results are again compared with an analogous statistical shape analysis framework utilizing POD in lieu of ICA. The ICA approach is shown to produce shape features for the RVES that capture more localized variations in the shape across the set compared to the POD approach, and overall, the ICA approach produces features that represent the RVES variation throughout the set in a considerably different manner than the more traditional POD approach, providing a potentially useful alternate to statistically analyze such a set of shapes.

Comparison of a Modal Method and a Proper Orthogonal Decomposition approach for multi-group time-dependent reactor spatial kinetics

In this paper, two modelling approaches based on a Modal Method (MM) and on the Proper Orthogonal Decomposition (POD) technique, for developing a control-oriented model of nuclear reactor spatial kinetics, are presented and compared. Both these methods allow developing neutronics description by means of a set of ordinary differential equations. The comparison of the outcomes provided by the two approaches focuses on the capability of evaluating the reactivity and the neutron flux shape in different reactor configurations, with reference to a TRIGA Mark II reactor. The results given by the POD-based approach are higher-fidelity with respect to the reference solution than those computed according to the MM-based approach, in particular when the perturbation concerns a reduced region of the core. If the perturbation is homogeneous throughout the core, the two approaches allow obtaining comparable accuracy results on the quantities of interest. As far as the computational burden is concerned, the POD approach ensures a better efficiency rather than direct Modal Method, thanks to the ability of performing a longer computation in the preprocessing that leads to a faster evaluation during the on-line phase.

Absorption line black body distribution function evaluated with proper orthogonal decomposition for mixture of CO2 and H2O

Purpose – The purpose of this paper is to present a new approach of evaluation of the absorption line black body distribution function (ALBDF) for a mixture of gases. Currently published correlations, which are used to reproduce the ALBDF, treat only single gases. Design/methodology/approach – A discrete form of the ALBDF is generated using line by line (LBL) calculations. The latest spectroscopic database HITEMP 2010 is used for the generation of the absorption coefficient histogram, which is cumulated later in order to produce a tabulated form of the ALBDF. The proper orthogonal decomposition (POD) statistical method is employed for the reproduction of the ALBDF. Interpolation property of the POD allows to reproduce the ALBDF for arbitrary gas mixture parameters. Findings – POD proved to possess optimal interpolation properties. Results obtained by using POD are in very good agreement with LBL integration. Research limitations/implications – One have to be aware that the model generated with the POD method can be used only within the range of parameters used to build the model. The POD does not perform any property extrapolation. The model is limited to a mixture of two gases, namely CO2 and H2O. Expanding the number of gases used in the mixture may lead to a relatively large matrix system, which is difficult to process with the POD approach. Practical implications – The presented approach can be used to produce absorption coefficients values and their weights, which can be applied in the gas radiative properties description using the weighted sum of gray gas (WSGG) concept. The proposed model can be used with any radiative transfer equation solver which employs the WSGG approach. Originality/value – For the first time, radiative properties of gas mixtures are reproduced using the POD approach.

A model reduction approach for the variational estimation of vascular compliance by solving an inverse fluid–structure interaction problem

Scientific computing has progressively become an important tool for research in cardiovascular diseases. The role of quantitative analyses based on numerical simulations has moved from ‘proofs of concept’ to patient-specific investigations, thanks to a strong integration between imaging and computational tools. However, beyond individual geometries, numerical models require the knowledge of parameters that are barely retrieved from measurements, especially in vivo. For this reason, recently cardiovascular mathematics considered data assimilation procedures for extracting the knowledge of patient-specific parameters from measures and images. In this paper, we consider specifically the quantification of vascular compliance, i.e. the parameter quantifying the tendency of arterial walls to deform under blood stress. Following up a previous paper, where a variational data assimilation procedure was proposed, based on solving an inverse fluid–structure interaction problem, here we consider model reduction techniques based on a proper orthogonal decomposition approach to accomplish the solution of the inverse problem in a computationally efficient way.

Noise correction of turbulent spectra obtained from acoustic doppler velocimeters

Velocity spectra are essential in characterizing turbulent flows. The Acoustic Doppler Velocimeter (ADV) provides three-dimensional time series data at a single point in space which are used for calculating velocity spectra. However, ADV data are susceptible to contamination from various sources, including instrument noise, which is the intrinsic limit to the accuracy of acoustic Doppler processing. This contamination results in a flattening of the velocity spectra at high frequencies (O(10)Hz). This paper demonstrates two elementary methods for attenuating instrument noise and improving velocity spectra. First, a “Noise Auto-Correlation” (NAC) approach utilizes the correlation and spectral properties of instrument noise to identify and attenuate the noise in the spectra. Second, a Proper Orthogonal Decomposition (POD) approach utilizes a modal decomposition of the data and attenuates the instrument noise by neglecting the higher-order modes in a time-series reconstruction. The methods are applied to ADV data collected in a tidal channel with maximum horizontal mean currents up to 2m/s. The spectra estimated using both approaches exhibit an f−5/3 slope, consistent with a turbulent inertial sub-range, over a wider frequency range than the raw spectra. In contrast, a Gaussian filter approach yields spectra with a sharp decrease at high frequencies. In an example application, the extended inertial sub-range from the NAC method increased the confidence in estimating the turbulent dissipation rate, which requires fitting the amplitude of the f−5/3 region. The resulting dissipation rates have smaller uncertainties and are more consistent with an assumed local balance to shear production, especially for mean horizontal currents less than 0.8m/s.

POD-based modeling on thermal-flow characteristics for air-cooled condenser of power plant

In order to rapidly and accurately predict the velocity and temperature distributions in air cooled condenser (ACC) of direct air cooling power plant influenced by the ambient wind, reduced order models (ROMs) for two-dimensional flow and temperature fields were studied with proper orthogonal decomposition (POD) method. The velocity vector, u, turbulent kinetic energy, k, and turbulent dissipation rate, ε, were approximated through POD procedure. The improved complementary POD approach (PODc) was adopted to construct robust velocity POD models of ACC under various ambient wind velocities, then flux matching procedure (FMP) was introduced to estimate the POD mode coefficients. And that of k, ε were obtained through cubic spline interpolation. It was found that as the case differed from the system reference point was great, the advantage of PODc became obvious. Otherwise, the mean relative errors of POD solutions were smaller than that of PODc method for the cases near system reference point. By interpolating the above three variables obtained by POD/PODc method into CFD software for energy equation, the temperature distributions could be acquired with the mean relative error of 10−3–10−4. At the meanwhile, the computing time spending on the energy equation iterating could be decreased significantly compared with that of direct CFD simulations.

Physically based regularization of hydrogeophysical inverse problems for improved imaging of process-driven systems

[1] We introduce a new strategy for integrating hydrologic process information as a constraint within hydrogeophysical imaging problems. The approach uses a basis-constrained inversion where basis vectors are tuned to the hydrologic problem of interest. Tuning is achieved using proper orthogonal decomposition (POD) to extract an optimal basis from synthetic training data generated by Monte Carlo simulations representative of hydrologic processes at a site. A synthetic case study illustrates that the approach performs well relative to other common inversion strategies for imaging a solute plume using an electrical resistivity survey, even when the initial conceptualization of hydrologic processes is incorrect. In two synthetic case studies, we found that the POD approach was able to significantly improve imaging of the plume by reducing the root mean square error of the concentration estimates by a factor of two. More importantly, the POD approach was able to better capture the bimodal nature of the plume in the second case study, even though the prior conceptual model for the POD basis was for a single plume. The ability of the POD inversion to improve concentration estimates exemplifies the importance of integrating process information within geophysical imaging problems. In contrast, the ability to capture the bimodality of the plume in the second example indicates the flexibility of the technique to move away from this prior process constraint when it is inconsistent with the observed ERI data.

Modification to adaptive model reduction for regulation of distributed parameter systems with fast transients

We focus on output feedback control of distributed processes whose infinite dimensional representation in appropriate Hilbert subspaces can be decomposed to finite dimensional slow and infinite dimensional fast subsystems. The controller synthesis issue is addressed using a refined adaptive proper orthogonal decomposition (APOD) approach to recursively construct accurate low dimensional reduced order models (ROMs) based on which we subsequently construct and couple almost globally valid dynamic observers with robust controllers. The novelty lies in modifying the data ensemble revision approach within APOD to enlarge the ROM region of attraction. The proposed control approach is successfully used to regulate the Kuramoto‐Sivashinsky equation at a desired steady state profile in the absence and presence of uncertainty when the unforced process exhibits nonlinear behavior with fast transients. The original and the modified APOD approaches are compared in different conditions and the advantages of the modified approach are presented. © 2013 American Institute of Chemical Engineers AIChE J, 59: 4595–4611, 2013

Transient Pull-In Analysis of Micro-Plate with Fluid-Structure Interactions

Electrostatic micro-devices are simple but important for MEMS applications. Precise dynamic descriptions of these devices are often hard to obtain due to the electrostatic nonlinearity and the fluid-structure interactions in devices. Here we present a comprehensive electrostatic-mechanical-fluidic coupling transient analysis for the pull-in process of two ends fixed micro-plate device. The numerical results are compared with the published experiment works of other researchers available in the literature, and thus the model had been validated. After that the proper orthogonal decomposition approach is performed for the snapshot matrixes which are sampled from an ensemble of the fully finite element results. The resulted spatial distribution modes of pressure show a higher spatial frequency toward the middle of the micro-plate, which indicates that the pressures at the moving edges of the plate are not equal to ambient pressure. Due to the increasing demands for simulation accuracy, the electrostatic-mechanical interactions and the nonlinear features of viscous loss from the surrounding fluid have to be taken into account in details.

The role of coherent turbulent structures in explaining scalar dissimilarity within the canopy sublayer

Scalar similarity is widely assumed in models and interpretation of micro-meteorological measurements. However, in the air space within and just above the canopy (the so-called canopy sublayer, CSL) scalar similarity is generally violated. The scalar dissimilarity has been mainly attributed to differences in the distribution of sources and sinks throughout the canopy. Since large-scale coherent structures in the CSL (e.g. double roller and sweep/ejection) arise from the instabilities generated by the interaction between the mean flow and the canopy, they may encode key dynamical features about the production term responsible for the source–sink dissimilarity of scalars. Therefore, it is reasonable to assume that the geometric attributes of coherent structures are tightly coupled to the onset and the vertical extent of scalar dissimilarity within the CSL. Large-eddy simulation (LES) runs were used to investigate the role of coherent structures in explaining scalar dissimilarity among three scalars (potential air temperature, water vapour and \(\text{ CO }_2\) concentration) within the CSL under near-neutral conditions for horizontally uniform but vertically varying vegetation leaf area density. It was shown that coherent structures, when identified from the first mode of a novel proper orthogonal decomposition (POD) approach, were able to capture some features of the scalar dissimilarity in the original LES field. This skill was quantified by calculating scalar–scalar correlation coefficients and turbulent Schmidt numbers of the original field and the coherent structures, respectively. However, coherent structures tend to magnify the magnitude of scalar–scalar correlation, particularly in cases where this correlation is already strong. The ability of coherent structures to describe more complex features such as the scalar sweep-ejection cycle was also explored. It was shown that the first mode of the POD does not capture the relative importance of sweeps to ejections in the original LES field. However, the superposition of few secondary coherent structures, derived from higher order POD modes, largely diminish the discrepancies between the original field and the POD expansion.

Wave Equation Simulation Using a Compressed Modeler

Repeated simulations of large scale wave propagation problems are prevalent in many fields. In oil exploration earth imaging problems, the use of full wave simulations is becoming routine and it is only hampered by the extreme computational resources required. In this contribution, we explore the feasibility of employing reduced-order modeling techniques in an attempt to significantly decrease the cost of these calculations. We consider the acoustic wave equation in two-dimensions for simplicity, but the extension to three-dimensions and to elastic or even anysotropic problems is clear. We use the proper orthogonal decomposition approach to model order reduction and describe two algorithms: the traditional one using the SVD of the matrix of snapshots and a more economical and flexible one using a progressive QR decomposition. We include also two a posteriori error estimation procedures and extensive testing and validation is presented that indicates the promise of the approach.

Improved extrapolation of steady turbulent aerodynamics using a non-linear POD-based reduced order model

AbstractA reduced-order modelling (ROM) approach for predicting steady, turbulent aerodynamic flows based on computational fluid dynamics (CFD) and proper orthogonal decomposition (POD) is presented. Model-order reduction is achieved by parameter space sampling, solution space representation via POD and restriction of a CFD solver to the POD subspace. Solving the governing equations of fluid dynamics is replaced by solving a non-linear least-squares optimisation problem. The method will be referred to as LSQ-ROM method. Two approaches of extracting POD basis information from CFD snapshot data are discussed: POD of the full state vector (global POD) and POD of each of the partial states separately (variable-by-variable POD). The method at hand is demonstrated for a 2D aerofoil (NACA 64A010) as well as for a complete industrial aircraft configuration (NASA Common Research Model) in the transonic flow regime by computing ROMs of the compressible Reynolds-averaged Navier-Stokes equations, pursuing both the global and the variable-by-variable POD approach. The LSQ-ROM approach is tried for extrapolatory flow conditions. Results are juxtaposed with those obtained by POD-based extrapolation using Kriging and the radial basis functions spline method. As a reference, the full-order CFD solutions are considered. For the industrial aircraft configuration, the cost of computing the reduced-order solution is shown to be two orders of magnitude lower than that of computing the reference CFD solution.

Bayesian Identification of Elastic Constants in Multi-Directional Laminate from Moiré Interferometry Displacement Fields

The ply elastic constants needed for classical lamination theory analysis of multi-directional laminates may differ from those obtained from unidirectional laminates because of three dimensional effects. In addition, the unidirectional laminates may not be available for testing. In such cases, full-field displacement measurements offer the potential of identifying several material properties simultaneously. For that, it is desirable to create complex displacement fields that are strongly influenced by all the elastic constants. In this work, we explore the potential of using a laminated plate with an open-hole under traction loading to achieve that and identify all four ply elastic constants (E 1 , E 2 , ν 12 , G 12 ) at once. However, the accuracy of the identified properties may not be as good as properties measured from individual tests due to the complexity of the experiment, the relative insensitivity of the measured quantities to some of the properties and the various possible sources of uncertainty. It is thus important to quantify the uncertainty (or confidence) with which these properties are identified. Here, Bayesian identification is used for this purpose, because it can readily model all the uncertainties in the analysis and measurements, and because it provides the full coupled probability distribution of the identified material properties. In addition, it offers the potential to combine properties identified based on substantially different experiments. The full-field measurement is obtained by moiré interferometry. For computational efficiency the Bayesian approach was applied to a proper orthogonal decomposition (POD) of the displacement fields. The analysis showed that the four orthotropic elastic constants are determined with quite different confidence levels as well as with significant correlation. Comparison with manufacturing specifications showed substantial difference in one constant, and this conclusion agreed with earlier measurement of that constant by a traditional four-point bending test. It is possible that the POD approach did not take full advantage of the copious data provided by the full field measurements, and for that reason that data is provided for others to use (as on line material attached to the article).

Galerkin approximation with proper orthogonal decomposition : new error estimates and illustrative examples

We propose a numerical analysis of proper orthogonal decomposition (POD) model reductions in which a priori error estimates are expressed in terms of the projection errors that are controlled in the construction of POD bases. These error estimates are derived for generic parabolic evolution PDEs, including with non-linear Lipschitz right-hand sides, and for wave-like equations. A specific projection continuity norm appears in the estimates and – whereas a general uniform continuity bound seems out of reach – we prove that such a bound holds in a variety of Galerkin bases choices. Furthermore, we directly numerically assess this bound – and the effectiveness of the POD approach altogether – for test problems of the type considered in the numerical analysis, and also for more complex equations. Namely, the numerical assessment includes a parabolic equation with super-linear reaction terms, inspired from the FitzHugh-Nagumo electrophysiology model, and a 3D biomechanical heart model. This shows that the effectiveness established for the simpler models is also achieved in the reduced-order simulation of these highly complex systems.

Feedback control of laminar flow behind backward-facing step by POD analysis and using perturbed Navier–Stokes equations

The purpose of this article is to design a reduced order model based on the proper orthogonal decomposition/Galerkin projection and perturbation method to develop a non-autonomous model. The resulting model can be used in optimal control of flow over backward-facing step. The main disadvantage of the proper orthogonal decomposition approach for control purposes is that, controlling parameters or inputs do not show up explicitly in the resulting reduced order system. The perturbation method can solve this problem and insert control inputs in the resulting system. The resulting system captures the time-varying influence of the controlling parameters and precisely predicts the Navier–Stokes response to external excitations. At last, optimal control theory is introduced to design a control law for a non-linear forced reduced model, which attempts to minimize the vorticity content in the fluid domain. The test bed is laminar flow behind backward-facing step [Formula: see text] actuated by a pair of blowing/suction jets. Results show that the wall jet can significantly influence the flow field and delay separation, while the perturbation method can predict the flow field in an accurate manner. The method is also found to be fast and efficient in computational time.

Filtered POD‐based low‐dimensional modeling of the 3D turbulent flow behind a circular cylinder

AbstractLow‐dimensional models have proven essential for feedback control and estimation of flow fields. While feedback control based on global flow estimation can be very efficient, it is often difficult to estimate the flow state if structures of very different length scales are present in the flow. The conventional snapshot‐based proper orthogonal decomposition (POD), a popular method for low‐order modeling, does not separate the structures according to size, since it optimizes modes based on energy. Two methods are developed in this study to separate the structures in the flow based on size. One of them is Hybrid Filtered POD method and the second one is 3D FFT‐based Filtered POD approach performed using a fast Fourier transform (FFT)‐based spatial filtering. In both the methods, a spatial low‐pass filter is employed to precondition snapshot sets before deriving POD modes. Three‐dimensional flow data from the simulation of turbulent flow over a circular cylinder wake at Re=20000 is used to evaluate the performance of the two methods. Results show that both the FFT‐based 3D Filtered POD and Hybrid Filtered POD are able to capture the large‐scale features of the flow, such as the von Karman vortex street, while not being contaminated by small‐scale turbulent structures present in the flow. Copyright © 2010 John Wiley & Sons, Ltd.

A dual-weighted trust-region adaptive POD 4D-VAR applied to a finite-element shallow-water equations model

We consider a limited-area finite-element discretization of the shallow-water equations model. Our purpose in this paper is to solve an inverse problem for the above model controlling its initial conditions in presence of observations being assimilated in a time interval (window of assimilation). We then attempt to obtain a reduced-order model of the above inverse problem, based on proper orthogonal decomposition (POD), referred to as POD 4-D VAR. Different approaches of POD implementation of the reduced inverse problem are compared, including a dual-weighed method for snapshot selection coupled with a trust-region POD approach. Numerical results obtained point to an improved accuracy in all metrics tested when dual-weighing choice of snapshots is combined with POD adaptivity of the trust-region type. Results of ad-hoc adaptivity of the POD 4-D VAR turn out to yield less accurate results than trust-region POD when compared with high-fidelity model. Directions of future research are finally outlined. Copyright © 2009 John Wiley & Sons, Ltd.