Proper Orthogonal Decomposition Technique Research Articles

A reduced-order extrapolation algorithm based on SFVE method and POD technique for non-stationary Stokes equations

In this paper, a reduced-order extrapolation algorithm (ROEA) based on proper orthogonal decomposition (POD) technique and classical stabilized finite volume element (SFVE) method for non-stationary Stokes equations is established. The error estimates between the ROEA solutions and the classical SFVE solutions and the implementation for solving ROEA are provided. Some numerical examples are used to verify that the results of numerical computation are consistent with theoretical conclusions. Moreover, it is shown that ROEA based on SFVE formulation and POD method is feasible and efficient for solving non-stationary Stokes equations.

Reinforcement learning based controller synthesis for flexible aircraft wings

Aeroelastic study of flight vehicles has been a subject of great interest and research in the last several years. Aileron reversal and flutter related problems are due in part to the elasticity of a typical airplane. Structural dynamics of an aircraft wing due to its aeroelastic nature are characterized by partial differential equations. Controller design for these systems is very complex as compared to lumped parameter systems defined by ordinary differential equations. In this paper, a stabilizing statefeedback controller design approach is presented for the heave dynamics of a wing-fuselage model. In this study, a continuous actuator in the spatial domain is assumed. A control methodology is developed by combining the technique of “proper orthogonal decomposition” and approximate dynamic programming. The proper orthogonal decomposition technique is used to obtain a low-order nonlinear lumped parameter model of the infinite dimensional system. Then a near optimal controller is designed using the single-network-adaptive-critic technique. Furthermore, to add robustness to the nominal single-network-adaptive-critic controller against matched uncertainties, an identifier based adaptive controller is proposed. Simulation results demonstrate the effectiveness of the single-network-adaptive-critic controller augmented with adaptive controller for infinite dimensional systems.

Application of low-dimensional finite element method to fractional diffusion equation

Classical finite element method (FEM) has been applied to solve some fractional differential equations, but its scheme has too many degrees of freedom. In this paper, a low-dimensional FEM, whose number of basis functions is reduced by the theory of proper orthogonal decomposition (POD) technique, is proposed for the time fractional diffusion equation in two-dimensional space. The presented method has the properties of low dimensions and high accuracy so that the amount of computation is decreased and the calculation time is saved. Moreover, error estimation of the method is obtained. Numerical example is given to illustrate the feasibility and validity of the low-dimensional FEM in comparison with traditional FEM for the time fractional differential equations.

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.

Analysis of Dynamic Stall Using Dynamic Mode Decomposition Technique

Dynamic mode decomposition is applied to investigate the unsteady flowfield around a pitching airfoil. The extracted flow structures are termed as dynamic mode decomposition modes. Analyses are performed for both attached flow and dynamic stall cases. Initially, flowfield data generated from numerical simulations are investigated. The effect of exclusion of the flowfield near the surface of the airfoil on the structure of the dynamic mode decomposition modes is examined. This is of vital importance for the experimental measurements, as the flowfield near the airfoil’s surface is difficult to measure using particle image velocimetry. In the latter part of the paper, dynamic mode decomposition modes extracted from the experimental data are analyzed. The structure of these modes are compared with the modes obtained from the proper orthogonal decomposition technique. Finally, effects of phase averaging on the modes are discussed.

Analyzing the fluctuating pressures acting on a circular cylinder using stochastic decomposition

Fluctuating wind pressures acting on bluff bodies are influenced by approaching turbulence and signature (body-induced) turbulence. For a circular cylinder, the signature turbulence is closely related to the formation of Karman vortex shedding. In this paper, proper orthogonal decomposition (POD) and spectral proper transformation techniques (SPT) are applied to the pressure fluctuations acting on a circular cylinder. The physical relationships between the decomposed modes and vortex shedding are discussed to identify the dominant aerodynamic behavior (lift or drag) and to evaluate its contribution to overall behavior. The effect of Reynolds number (Re) is also addressed. It is found that the application of POD and SPT can separate the along-wind and across-wind effects on the cylinder model in both subcritical and supercritical regimes. In contrast to POD, the SPT mode is formulated in the frequency domain, and the dynamic coherent structures can be defined in terms of amplitude and phase angle, which allows detection of the advection features of vortex shedding. In addition, it is observed that the energy contribution of the shedding induced lift force increases with Re and gradually becomes a dominant aerodynamic force at Reynolds numbers in the supercritical regime.

On the sensitivity of the POD technique for a parameterized quasi-nonlinear parabolic equation

In what follows, we consider the Proper Orthogonal Decomposition (POD) technique of model order reduction, for a parameterized quasi-nonlinear parabolic equation. A POD basis associated with a set of reference values of the characteristic parameters is considered. From this basis, a parametric reduced order model (ROM) projecting the initial equation is constructed. A mathematical a priori estimate of the parametric squared L2-error induced by this projection is developed. This later estimate is based on both, the parametric behavior of the squared L2-ROM-error thanks to the resolution of a Ricatti differential inequality in the parametric ROM-error, and the convergence rate of the parametric ROM to the full problem, via the augmentation of the basis dimension. Indeed, under restrictive conditions on the solutions regularity of such equations, we are able to precise the slope of the logarithm of the squared L2-norm of the ROM error, as a function of the logarithm of the basis modes number. Numerical experiments of our theoretical estimate, are presented for the 2D Navier-Stokes equations in the case of an unsteady and incompressible fluid flow in a channel around a circular cylinder. A mathematical a priori estimate of the parametric squared L2-error induced by the model reduction by POD is developped for a parameterized quasi-nonlinear parabolic equation. This estimate is obtained thanks to the resolution of a Ricatti differential inequality.

PIV Measurements and POD Analysis during Natural Convection with Protruded Heater in a Rectangular Enclosure

An extensive measurement using 2-D Particle Image Velocimetry (PIV) technique has been carried out during natural convection with a heating sources protruded on the non-heated lower surface of a rectangular enclosure. A time-resolved data series have been captured to analyse the effect of different Rayleigh number on flow fields for fluid of Prandtl number 0.71. A fast and efficient data analysis tool based on the Proper Orthogonal Decomposition (POD) technique has been used to create low dimensional approximations of high dimensional systems for predicting the dominant flow structures. This includes interpolating results at off-design parameters, along with knowledge about contours of different higher modes as well as obtaining information about the energy content in each specific mode. The mean non fluctuating mode showed maximum energy and the same decreased with increasing mode number. This allowed a clear detailed understanding about the complex non-linearities in the problem using a linear decomposition technique

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.

Application of the smooth orthogonal decomposition to oceanographic data sets

The so-called “smooth orthogonal decomposition” technique, developed in the nonlinear vibrations and fatigue community, is applied to an oceanographic data set. This decomposition technique overcomes some limitations of the proper orthogonal decomposition technique by identifying modes which behave smoothly in time and thus being sensitive to both variance amplitude as well as frequency. This important property is demonstrated through identification of a 4 day topographic Rossby wave on the Louisiana slope, which the proper orthogonal decomposition technique fails to isolate.

Aerodynamic Stabilization Mechanism of a Twin Box Girder with Various Slot Widths

Five representative girder cross sections with various slot widths are utilized to analyze the effects of center slots on their aerodynamic performance, based on wind-tunnel tests and theoretical analyses. It is shown that the favorable aerodynamic effects of the center slot on bridge decks depend on the aerodynamic shape of the box girders and on the slot widths rather than unconditionally improving the aeroelastic stability. Further investigation of a streamlined box girder with various slot widths results in a modified Selberg formula to calculate the critical flutter wind speed for design purposes, wherein the Lorentz peak-value function is utilized. The flutter mechanism is illustrated utilizing a two-dimensional three-degrees-of-freedom (2D-3DOF) analysis scheme. The results indicate that the center slot changes the participation level of the heaving motion at the flutter onset, which is highly correlated with the critical flutter wind speed. In addition, particle image velocimetry (PIV) and proper orthogonal decomposition (POD) techniques are employed to assist in revealing the aerodynamic stabilization mechanism of the center slotting of box girders.

A reduced-order extrapolation algorithm based on CNLSMFE formulation and POD technique for two-dimensional Sobolev equations

A reduced-order extrapolation algorithm based on Crank-Nicolson least-squares mixed finite element (CNLSMFE) formulation and proper orthogonal decomposition (POD) technique for two-dimensional (2D) Sobolev equations is established. The error estimates of the reduced-order CNLSMFE solutions and the implementation for the reduced-order extrapolation algorithm are provided. A numerical example is used to show that the results of numerical computations are consistent with theoretical conclusions. Moreover, it is shown that the reduced-order extrapolation algorithm is feasible and efficient for seeking numerical solutions to 2D Sobolev equations.

On the Hybrid Combustion Instability

In the present paper, a combined method of large eddy simulations for non-premixed combustion in a turbulent flow coupled with proper orthogonal decomposition of instantaneous velocity, pressure and temperature fields is developed in order to identify the effect of coherent structure and to obtain a reduced order model for control model. First we investigate the reacting flow using Large Eddy Simulations technique. This physical model is pertinent to internal flows inside the hybrid rocket motors. The turbulence-combustion interaction is based on a combination of finite rate/eddy dissipation model applied to a reduced chemical mechanism with four reactions. Next, the paper refers to the derivation of a Reduced Order Model (ROM) for the same problem, based on the Proper Orthogonal Decomposition (POD) technique. ROMs are used to obtain fast and accurate results, needed in the areas of flow control. The flow and thermal fields obtained with ROMs are compared with the ones obtained from the full simulation and an analysis on the number of modes required to achieve the desired accuracy is presented. Finally, a static control technique is proposed.

An Overview of Unsteady Analysis Techniques for Natural Wind Turbulence and its Effects on Natural Ventilation

Focusing on the turbulence in natural ventilation and its impact on both occupant thermal comfort and building energy consumption, this paper presents a review of existing unsteady natural ventilation envelope flow models, as well as other techniques that have potential application to further our understanding of turbulence in natural ventilation and develop models which capture the dynamics and effects on thermal comfort. Literature provides numerous techniques ranging from quasi-steady temporal inertial theory, to unsteady CFD models, to experimental study, and other statistical and signal processing techniques. The Proper Orthogonal Decomposition and Stochastic Estimate techniques have been paid particular attention as they may be an alternative tool for understanding the turbulence of natural wind and possibly be combined with predictive models in an attempt to quantify the effects on thermal comfort and/or optimization of building energy management system’s operation.

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.

Model order reduction for Bayesian approach to inverse problems

This work presents an approach to solve inverse problems in the application of water quality management in reservoir systems. One such application is contaminant cleanup, which is challenging because tasks such as inferring the contaminant location and its distribution require large computational efforts and data storage requirements. In addition, real systems contain uncertain parameters such as wind velocity; these uncertainties must be accounted for in the inference problem. The approach developed here uses the combination of a reduced-order model and a Bayesian inference formulation to rapidly determine contaminant locations given sparse measurements of contaminant concentration. The system is modelled by the coupled Navier-Stokes equations and convection-diffusion transport equations. The Galerkin finite element method provides an approximate numerical solution-the ’full model’, which cannot be solved in real-time. The proper orthogonal decomposition and Galerkin projection technique are applied to obtain a reduced-order model that approximates the full model. The Bayesian formulation of the inverse problem is solved using a Markov chain Monte Carlo method for a variety of source locations in the domain. Numerical results show that applying the reduced-order model to the source inversion problem yields a speed-up in computational time by a factor of approximately 32 with acceptable accuracy in comparison with the full model. Application of the inference strategy shows the potential effectiveness of this computational modeling approach for managing water quality.

The Estimation of Regional Crop Yield Using Ensemble-Based Four-Dimensional Variational Data Assimilation

To improve crop model performance for regional crop yield estimates, a new four-dimensional variational algorithm (POD4DVar) merging the Monte Carlo and proper orthogonal decomposition techniques was introduced to develop a data assimilation strategy using the Crop Environment Resource Synthesis (CERES)-Wheat model. Two winter wheat yield estimation procedures were conducted on a field plot and regional scale to test the feasibility and potential of the POD4DVar-based strategy. Winter wheat yield forecasts for the field plots showed a coefficient of determination (R2) of 0.73, a root mean square error (RMSE) of 319 kg/ha, and a relative error (RE) of 3.49%. An acceptable yield at the regional scale was estimated with an R2 of 0.997, RMSE of 7346 tons, and RE of 3.81%. The POD4DVar-based strategy was more accurate and efficient than the EnKF-based strategy. In addition to crop yield, other critical crop variables such as the biomass, harvest index, evapotranspiration, and soil organic carbon may also be estimated. The present study thus introduces a promising approach for operationally monitoring regional crop growth and predicting yield. Successful application of this assimilation model at regional scales must focus on uncertainties derived from the crop model, model inputs, data assimilation algorithm, and assimilated observations.

Model reduction for reacting flow applications

A model reduction approach based on Galerkin projection, proper orthogonal decomposition (POD), and the discrete empirical interpolation method (DEIM) is developed for chemically reacting flow applications. Such applications are challenging for model reduction due to the strong coupling between fluid dynamics and chemical kinetics, a wide range of temporal and spatial scales, highly nonlinear chemical kinetics, and long simulation run-times. In our approach, the POD technique combined with Galerkin projection reduces the dimension of the state (unknown chemical concentrations over the spatial domain), while the DEIM approximates the nonlinear chemical source term. The combined method provides an efficient offline–online solution strategy that enables rapid solution of the reduced-order models. Application of the approach to an ignition model of a premixed H2/O2/Ar mixture with 19 reversible chemical reactions and 9 species leads to reduced-order models with state dimension several orders of magnitude smaller than the original system. For example, a reduced-order model with state dimension of 60 accurately approximates a full model with a dimension of 91,809. This accelerates the simulation of the chemical kinetics by more than two orders of magnitude. When combined with the full-order flow solver, this results in a reduction of the overall computational time by a factor of approximately 10. The reduced-order models are used to analyse the sensitivity of outputs of interest with respect to uncertain input parameters describing the reaction kinetics.

Experimental investigation on laminar separation control for flow over a two-dimensional bump

The laminar boundary layer separation flow over a two-dimensional bump controlled by synthetic jets is experimentally investigated in a water channel with hydrogen-bubble visualisation and particle image velocimetry (PIV) techniques. The two-dimensional synthetic jet is applied near the separation point. Two Reynolds numbers (Re = 700 and 1120) based on the bump height and free-stream velocity are adopted in this experiment, and seven different excitation frequencies at each Reynolds number are considered, focusing on the separation control as well as the vortex dynamics. The experimental results show that the optimal control can only be achieved within some excitation frequencies at both Reynolds numbers. However, beyond this range, further increasing the excitation frequency leads to an increase in the separation region. The proper orthogonal decomposition (POD) technique and vortex identification by swirling strength (Λci) are applied for the deeper analysis of the separated flow. The reconstructed Λci field by the first four POD modes is used and vortex lock-on phenomenon is observed. It is found that the negative synthetic jet vortex with clockwise rotation draws the separated wake shear layer as it is convected downstream, and then they syncretise together. Thus, the new vortex is induced and shedding downstream periodically.

Load field reconstruction with a combined POD and integral spline approximation technique

In this paper, a new technique for determining a load field (e.g., pressure) on the basis of a few global measurements (e.g., forces) is presented. This technique is based on a combination of proper orthogonal decomposition (POD) and polynomial spline approximation with integral constraints and is here illustrated with respect to the prediction of the wave load distribution along a slender floating body like a ship. The input data are provided by the time-histories of the lumped vertical forces acting on several longitudinal portions (i.e., segments) of a scaled model of a fast ship. To achieve a better understanding of the accuracy of this technique, the segment forces are not experimentally but numerically obtained by integrating the sectional forces over the length of the segments. These forces, calculated with a strip-theory approach precessing experimental data relative to ship motion in waves, provide also the target distribution for validating the present procedure. The set of lumped hydrodynamic forces defines the vector process to which POD is applied. The components of the identified POD modes provide the integral constraints for the spline polynomials approximating the continuous basis functions of the sectional load distribution. The sectional load distribution along the hull is then reconstructed, showing good agreement with the original load data. Finally, the robustness of the proposed technique is investigated by studying the propagation of bias errors and additive white noise from input data to final results.