Application of proper orthogonal decomposition to the discrete Euler equations

Pettit, Chris L.; Beran, Philip S.

doi:10.1002/nme.510

Cited by 34 publications

(11 citation statements)

References 23 publications

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“…2, because the eigenvectors are not functions of x [19]. Details for mastering the art of POD method are published widely and readers can follow the links here for further readings [20][21][22][23].…”

Section: The Pod Methodsmentioning

confidence: 99%

Development and application of reduced‐order neural network model based on proper orthogonal decomposition for BOD5 monitoring: Active and online prediction

Noori

Karbassi

Ashrafi

et al. 2011

Env Prog and Sustain Energy

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This study aims at providing a reduced‐order neural network model (RONNM) based on proper orthogonal decomposition (POD) for online prediction of the 5 days biochemical oxygen demand (BOD5). To achieve this goal, the POD method is used to reduce the input vectors to neural network (NN) model. Evaluation of the selected RONNM and its comparison with the model fed with all input vectors (NN model) shows that it reduces not only the output error but also computational cost. Results depict that Pearson correlation coefficient (R) and root mean square error for the best fitted RONNM are 0.94 and 7.75, respectively. Furthermore, accuracy analysis of the output results of models based on developed discrepancy ratio statistic reveals that RONNM is superior. Thus, developed methodology can be considered as an effective tool to train the water resources managers for online evaluation of water quality related to BOD5. © 2011 American Institute of Chemical Engineers Environ Prog, 32: 120–127, 2013.

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Section: The Pod Methodsmentioning

confidence: 99%

Development and application of reduced‐order neural network model based on proper orthogonal decomposition for BOD5 monitoring: Active and online prediction

Noori

Karbassi

Ashrafi

et al. 2011

Env Prog and Sustain Energy

View full text Add to dashboard Cite

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“…In our approach to computing POD modes, when there are multiple variables per grid point, each mode corresponds to one of the field variables [18]. The modal content associated with other variables is specified to vanish.…”

Section: The Proper Orthogonal Decomposition (Pod)mentioning

confidence: 99%

A Reduced Order Cyclic Method for Computation of Limit Cycles

Beran

Lucia

2005

Nonlinear Dyn

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A reduced order cyclic method was developed to compute limit-cycle oscillations for large, nonlinear, multidisciplinary systems of equations. Method efficacy was demonstrated for two simplified models: a typical-section airfoil with nonlinear structural coupling and a nonlinear panel in high-speed flow. The cyclic method was verified to maintain second-order temporal accuracy, yield converged limit cycles in about 10 Newton iterates, and provide precise estimates of cycle frequency. This method was projected onto a low-order space using a set of variables governing the amplitudes of empirically derived modes, which were computed with the proper orthogonal decomposition. In this reduced order form, the cyclic Jacobian was greatly compressed, allowing accurate limit cycle solutions to be very efficiently computed. Nomenclature s = Time step normalized by period t = Time step x = Panel node spacing γ = Ratio of specific heats λ = Reduced velocity (airfoil); pressure parameter (panel) µ = Mass ratio = Mode matrix ν = Poisson's ratio, 0.3 M ∞ = Freestream Mach number N f = Number of variables N m = Number of modes N t = Number of time intervals in oscillation N x = Number of nodes on panel (excluding end points) p = Pressure T = Oscillation period w = Panel deflection v = Panel velocity, ∂w/∂t

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“…An approximate method suited to nonlinear systems is reduced-order modeling (ROM), which represents a full order model with an optimal basis, in the mean square sense, through a KarhunenLoeve expansion (KLE) [12][13][14][15]. ROM has the advantage that it can be efficiently tuned to capture flow physics at a high fidelity.…”

Section: Doi: 102514/111468mentioning

confidence: 99%

Uncertainty Quantification with a B-Spline Stochastic Projection

et al. 2006

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Presented is a new stochastic algorithm for computing the probability density functions and estimating bifurcations of nonlinear equations of motion whose system parameters are characterized by a Gaussian distribution. Polynomial and Fourier chaos expansions, which are spectral methods, have been used successfully to propagate parametric uncertainties in nonlinear systems. However, it is shown that bifurcations in the time domain are manifested as discontinuities in the stochastic domain, which are problematic for solution with these spectral approaches. Because of this, a new algorithm is introduced based on the stochastic projection method but employing a multivariate B spline. Samples are obtained by choosing nodes on the stochastic axes. These samples are used to build an interpolating function in the stochastic domain. Monte Carlo simulations are then very efficiently performed on this interpolating function to estimate probability density functions of a response. The results from this nonintrusive and non-Galerkin approach are in excellent agreement with Monte Carlo simulations of the governing equations, but at a computational cost 2 orders of magnitude less than a traditional Monte Carlo approach. The probability density functions obtained from the stochastic algorithm provide a rapid estimate of the probability of failure for a nonlinear pitch and plunge airfoil. Nomenclature a h = distance from the airfoil midchord to the elastic axis, positive aft B i;k 1 ;x 1 = the ith B spline of order k 1 for the knot sequence x 1 B i;k 2 ;x 2 = the jth B spline of order k 2 for the knot sequence= the second moment of inertia of the airfoil about the elastic axis, mr 2 b 2 K h = plunge stiffness of the airfoil, m! 2 h K = torsional stiffness of the airfoil about the elastic axis, I ! 2 k 1 , k 2 = order of the B splines on the 1 and 2 axes, respectively m = mass of the airfoil N 1 , N 2 = number of knots on the 1 and 2 axes, respectively S = first moment of inertia of the airfoil about the elastic axis, mx b t = nondimensional time, scaled by V 1 and b V r = reduced velocity, V 1 =! b V 1 = freestream velocity w = Gaussian weight function x = distance from the elastic axis to the center of mass of the airfoil, positive aft x 1 , x 2 = vectors containing the knots on the 1 and 2 axes, respectively = airfoil pitch = airfoil cubic spring constant = airfoil quintic spring constant s = airfoil mass ratio, m= 1 b 2 1 , 2 = zero mean, unit variance Gaussian random variables 1 = freestream density = random basis functions ! r = frequency ratio, ! h =! ! , ! h = uncoupled natural frequencies in pitch and plunge, respectively Superscripts = mean valuẽ = standard deviation = expansion coefficient

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Application of proper orthogonal decomposition to the discrete Euler equations

Cited by 34 publications

References 23 publications

Development and application of reduced‐order neural network model based on proper orthogonal decomposition for BOD5 monitoring: Active and online prediction

Development and application of reduced‐order neural network model based on proper orthogonal decomposition for BOD5 monitoring: Active and online prediction

A Reduced Order Cyclic Method for Computation of Limit Cycles

Uncertainty Quantification with a B-Spline Stochastic Projection

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