An Energy Closure Criterion for Model Reduction of a Kicked Euler–Bernoulli Beam

2020 ◽  
Vol 143 (4) ◽  
Author(s):  
Suparno Bhattacharyya ◽  
Joseph P. Cusumano
Keyword(s):  
Model Reduction ◽  
Mode Selection ◽  
Computational Cost ◽  
Impulsive Loading ◽  
Bernoulli Beam ◽  
Full System ◽  
Fixed Amount ◽  
Reduced Order ◽  
Proper Orthogonal ◽  
Euler Bernoulli Beam

Abstract Reduced order models (ROMs) can be simulated with lower computational cost while being more amenable to theoretical analysis. Here, we examine the performance of the proper orthogonal decomposition (POD), a data-driven model reduction technique. We show that the accuracy of ROMs obtained using POD depends on the type of data used and, more crucially, on the criterion used to select the number of proper orthogonal modes (POMs) used for the model. Simulations of a simply supported Euler–Bernoulli beam subjected to periodic impulsive loads are used to generate ROMs via POD, which are then simulated for comparison with the full system. We assess the accuracy of ROMs obtained using steady-state displacement, velocity, and strain fields, tuning the spatiotemporal localization of applied impulses to control the number of excited modes in, and hence the dimensionality of, the system’s response. We show that conventional variance-based mode selection leads to inaccurate models for sufficiently impulsive loading and that this poor performance is explained by the energy imbalance on the reduced subspace. Specifically, the subspace of POMs capturing a fixed amount (say, 99.9%) of the total variance underestimates the energy input and dissipated in the ROM, yielding inaccurate reduced-order simulations. This problem becomes more acute as the loading becomes more spatio-temporally localized (more impulsive). Thus, energy closure analysis provides an improved method for generating ROMs with energetics that properly reflect that of the full system, resulting in simulations that accurately represent the system’s true behavior.

The Importance of Energy Criteria for Selecting Modes in Reduced Order Modeling

2019 ◽  
Author(s):  
Suparno Bhattacharyya ◽  
Joseph P. Cusumano
Keyword(s):  
Mode Selection ◽  
Selection Criterion ◽  
Fixed Amount ◽  
Reduced Order ◽  
New Energy ◽  
Total Data ◽  
Effective Dimensionality ◽  
Spatiotemporal Localization ◽  
Proper Orthogonal ◽  
Euler Bernoulli Beam

Abstract We study the performance of the proper orthogonal decomposition when used for model reduction of an Euler-Bernoulli beam subjected to periodic impulses. We assess the accuracy of reduced order models (ROMs) obtained using steady-state displacement time series. The spatiotemporal localization of the applied impulses is tuned to control the number of excited modes in, and hence the effective dimensionality of, the system’s response. We find that when the impacts are significantly localized (i.e., are more impulsive), the conventional variance-based mode selection criterion can lead to inaccurate ROMs. We show that this arises when the reduced subspace capturing a fixed amount (say, 99.9%) of the total data variance underestimates the energy input and/or dissipated in the ROM, leading to energy imbalance. We thus propose a new energy closure criterion that provides an improved method for generating ROMs. The energetics of the resulting ROMs properly reflect those of the full system, and yield simulations that accurately represent the system’s true behavior.

Asymmetric Bifurcation of Initially Curved Nanobeam

2015 ◽  
Vol 82 (9) ◽  
Author(s):  
X. Chen ◽  
S. A. Meguid
Keyword(s):  
Symmetry Breaking ◽  
Degrees Of Freedom ◽  
Surface Elasticity ◽  
Beam Theory ◽  
Surface Effects ◽  
Beam Model ◽  
Bernoulli Beam ◽  
Reduced Order ◽  
Euler Bernoulli Beam ◽  
Bifurcation Behavior

In this paper, we investigate the asymmetric bifurcation behavior of an initially curved nanobeam accounting for Lorentz and electrostatic forces. The beam model was developed in the framework of Euler–Bernoulli beam theory, and the surface effects at the nanoscale were taken into account in the model by including the surface elasticity and the residual surface tension. Based on the Galerkin decomposition method, the model was simplified as two degrees of freedom reduced order model, from which the symmetry breaking criterion was derived. The results of our work reveal the significant surface effects on the symmetry breaking criterion for the considered nanobeam.

scholarly journals Empirical Reduced-Order Modeling for Boundary Feedback Flow Control

2008 ◽  
Vol 2008 ◽  
pp. 1-11 ◽  
Author(s):  
Seddik M. Djouadi ◽  
R. Chris Camphouse ◽  
James H. Myatt
Keyword(s):  
Model Reduction ◽  
Reduction Method ◽  
Linear Subspace ◽  
Balanced Truncation ◽  
Order Model ◽  
Reduced Order ◽  
Domain Identification ◽  
Controllability And Observability ◽  
Obstacle Geometry ◽  
Proper Orthogonal

This paper deals with the practical and theoretical implications of model reduction for aerodynamic flow-based control problems. Various aspects of model reduction are discussed that apply to partial differential equation- (PDE-) based models in general. Specifically, the proper orthogonal decomposition (POD) of a high dimension system as well as frequency domain identification methods are discussed for initial model construction. Projections on the POD basis give a nonlinear Galerkin model. Then, a model reduction method based on empirical balanced truncation is developed and applied to the Galerkin model. The rationale for doing so is that linear subspace approximations to exact submanifolds associated with nonlinear controllability and observability require only standard matrix manipulations utilizing simulation/experimental data. The proposed method uses a chirp signal as input to produce the output in the eigensystem realization algorithm (ERA). This method estimates the system's Markov parameters that accurately reproduce the output. Balanced truncation is used to show that model reduction is still effective on ERA produced approximated systems. The method is applied to a prototype convective flow on obstacle geometry. AnH∞feedback flow controller is designed based on the reduced model to achieve tracking and then applied to the full-order model with excellent performance.

Reduced Order Models of Viscous Flows in Turbomachinery Using Proper Orthogonal Decomposition

1999 ◽  
Author(s):  
Bogdan I. Epureanu ◽  
Earl H. Dowell ◽  
Kenneth C. Hall
Keyword(s):  
Proper Orthogonal Decomposition ◽  
Frequency Domain ◽  
Degrees Of Freedom ◽  
Orthogonal Decomposition ◽  
Full Potential ◽  
Reduced Order Models ◽  
Full System ◽  
Integral Boundary ◽  
Reduced Order ◽  
Proper Orthogonal

Abstract The proper orthogonal decomposition technique is applied in the frequency domain to obtain reduced order models (ROM) of the flow in a cascade of airfoils. The flow is described by a inviscid-viscous interaction model where the inviscid part is described by the full potential equation and the viscous part is described by an integral boundary layer model. The fully nonlinear steady flow is computed and the unsteady flow is linearized about the steady solution. A frequency domain model is constructed and validated showing to provide similar results when compared with previous computational and experimental data presented in the literature. A cascade of airfoils forming a slightly modified Tenth Standard Configuration is numerically investigated. We show that the ROMs with only 10 to 40 degrees of freedom predict accurately the unsteady response of the full system with approximately 10,000 degrees of freedom for the subsonic case. We also show that the ROMs with 15 to 75 degrees of freedom predict accurately the unsteady response of the full system with approximately 17, 500 degrees of freedom for the transonic case. The ROMs are shown to be accurate both for a broad range of reduced frequencies and a full spectrum of interblade phase angles.

MODEL REDUCTION FOR FLUIDS, USING BALANCED PROPER ORTHOGONAL DECOMPOSITION

2005 ◽  
Vol 15 (03) ◽  
pp. 997-1013 ◽  
Author(s):  
C. W. ROWLEY
Keyword(s):  
Proper Orthogonal Decomposition ◽  
Model Reduction ◽  
Projection Method ◽  
Computational Cost ◽  
Plane Channel ◽  
Orthogonal Decomposition ◽  
Inner Product ◽  
High Dimensional ◽  
Balanced Truncation ◽  
Proper Orthogonal

Many of the tools of dynamical systems and control theory have gone largely unused for fluids, because the governing equations are so dynamically complex, both high-dimensional and nonlinear. Model reduction involves finding low-dimensional models that approximate the full high-dimensional dynamics. This paper compares three different methods of model reduction: proper orthogonal decomposition (POD), balanced truncation, and a method called balanced POD. Balanced truncation produces better reduced-order models than POD, but is not computationally tractable for very large systems. Balanced POD is a tractable method for computing approximate balanced truncations, that has computational cost similar to that of POD. The method presented here is a variation of existing methods using empirical Gramians, and the main contributions of the present paper are a version of the method of snapshots that allows one to compute balancing transformations directly, without separate reduction of the Gramians; and an output projection method, which allows tractable computation even when the number of outputs is large. The output projection method requires minimal additional computation, and has a priori error bounds that can guide the choice of rank of the projection. Connections between POD and balanced truncation are also illuminated: in particular, balanced truncation may be viewed as POD of a particular dataset, using the observability Gramian as an inner product. The three methods are illustrated on a numerical example, the linearized flow in a plane channel.

scholarly journals Development of a Reduced Order Model for Fuel Burnup Analysis

Energies ◽  
2020 ◽  
Vol 13 (4) ◽  
pp. 890 ◽  
Author(s):  
Christian Castagna ◽  
Manuele Aufiero ◽  
Stefano Lorenzi ◽  
Guglielmo Lomonaco ◽  
Antonio Cammi
Keyword(s):  
Methodological Approach ◽  
Computational Cost ◽  
Reaction Rates ◽  
Reduced Order Model ◽  
Fuel Burnup ◽  
Order Model ◽  
Reduced Order ◽  
Solution Accuracy ◽  
Proper Orthogonal ◽  
Over Time

Fuel burnup analysis requires a high computational cost for full core calculations, due to the amount of the information processed for the total reaction rates in many burnup regions. Indeed, they reach the order of millions or more by a subdivision into radial and axial regions in a pin-by-pin description. In addition, if multi-physics approaches are adopted to consider the effects of temperature and density fields on fuel consumption, the computational load grows further. In this way, the need to find a compromise between computational cost and solution accuracy is a crucial issue in burnup analysis. To overcome this problem, the present work aims to develop a methodological approach to implement a Reduced Order Model (ROM), based on Proper Orthogonal Decomposition (POD), in fuel burnup analysis. We verify the approach on 4 years of burnup of the TMI-1 unit cell benchmark, by reconstructing fuel materials and burnup matrices over time with different levels of approximation. The results show that the modeling approach is able to reproduce reactivity and nuclide densities over time, where the accuracy increases with the number of basis functions employed.

A Reduced-Order Model for Turbomachinery Flows Using Proper Orthogonal Decomposition

2013 ◽  
Author(s):  
Thomas A. Brenner ◽  
Forrest L. Carpenter ◽  
Brian A. Freno ◽  
Paul G. A. Cizmas
Keyword(s):  
Proper Orthogonal Decomposition ◽  
Stokes Equations ◽  
Computational Cost ◽  
Wheel Speed ◽  
Orthogonal Decomposition ◽  
Reduced Order Model ◽  
Computational Time ◽  
Order Model ◽  
Reduced Order ◽  
Proper Orthogonal

This paper presents the development of a reduced-order model based on the proper orthogonal decomposition (POD) method. The POD method has been developed to predict turbomachinery flows modeled by the Reynolds-averaged Navier–Stokes equations. The purpose of using a POD-based reduced-order model is to decrease the computational cost of turbomachinery flows. The POD model has been tested for two configurations: a canonical channel with a bump case and the transonic NASA Rotor 67 case. The Rotor 67 case has been simulated at design wheel speed and at three off-design conditions: 70, 80, and 90% of the wheel speed. The results of the POD-based reduced-order model where in excellent agreement with the full-order model results. The computational time of the reduced-order model was approximately one order of magnitude smaller than that of the full-order model.

Verification of Nonlinear Proper Orthogonal Decomposition Reduced Order Modeling for BWR Fuel Assemblies

2013 ◽  
Author(s):  
Dennis P. Prill ◽  
Andreas G. Class
Keyword(s):  
Proper Orthogonal Decomposition ◽  
Power Distribution ◽  
Computational Cost ◽  
Orthogonal Decomposition ◽  
Low Flow ◽  
Reduced Order ◽  
Linear Character ◽  
Operation Conditions ◽  
Non Linear ◽  
Proper Orthogonal

Thermal-hydraulic coupling between power, flow rate and density, intensified by neutronics feedback are the main drivers of boiling water reactor (BWR) stability behavior. Studying potential power oscillations require focusing on BWR operation at high-power low-flow conditions interacting with unfavorable power distribution. Current design rules assure admissible operation conditions by exclusion regions determined by numerical calculations and analytical methods. Analyzing an exhaustive parameter space of the non-linear BWR system becomes feasible with methodologies based on reduced order models (ROMs) saving computational cost and improving the physical understanding. A general reduction technique is given by the proper orthogonal decomposition (POD). Model-specific options and aspects of the POD-ROM-methodology are considered. A first verification is illustrated by means of a chemical tubular reactor (TR) setup. Experimental and analytical results for natural convection in a closed circuit (NCC) [1, 2] serve as a second verification example. This setup shows a strongly non-linear character. The implemented model is validated by means of a linear stability map. Transient behavior of the NCC-POD-ROM can not only reproduce the input data but rather predict different states.

A New Closure Strategy for Proper Orthogonal Decomposition Reduced-Order Models

2012 ◽  
Vol 7 (3) ◽  
Author(s):  
Imran Akhtar ◽  
Zhu Wang ◽  
Jeff Borggaard ◽  
Traian Iliescu
Keyword(s):  
Proper Orthogonal Decomposition ◽  
Computational Cost ◽  
Viscosity Coefficient ◽  
Laminar Flows ◽  
Orthogonal Decomposition ◽  
Frobenius Norm ◽  
Reduced Order Models ◽  
Closure Model ◽  
Reduced Order ◽  
Proper Orthogonal

Proper orthogonal decomposition (POD) is one of the most significant reduced-order modeling (ROM) techniques in fluid mechanics. However, the application of POD based reduced-order models (POD-ROMs) is primarily limited to laminar flows due to the decay of physical accuracy. A few nonlinear closure models have been developed for improving the accuracy and stability of the POD-ROMs, which are generally computationally expensive. In this paper we propose a new closure strategy for POD-ROMs that is both accurate and effective. In the new closure model, the Frobenius norm of the Jacobian of the POD-ROM is introduced as the eddy viscosity coefficient. As a first step, the new method has been tested on a one-dimensional Burgers equation with a small dissipation coefficient ν=10-3. Numerical results show that the Jacobian based closure model greatly improves the physical accuracy of the POD-ROM, while maintaining a low computational cost.

Galerkin proper orthogonal decomposition-reduced order method (POD-ROM) for solving generalized Swift-Hohenberg equation

2019 ◽  
Vol 29 (8) ◽  
pp. 2642-2665 ◽  
Author(s):  
Mehdi Dehghan ◽  
Mostafa Abbaszadeh ◽  
Amirreza Khodadadian ◽  
Clemens Heitzinger
Keyword(s):  
Proper Orthogonal Decomposition ◽  
Discontinuous Galerkin ◽  
Computational Cost ◽  
Numerical Procedure ◽  
Orthogonal Decomposition ◽  
Content Type ◽  
Reduced Order ◽  
Cpu Time ◽  
Ldg Method ◽  
Proper Orthogonal

Purpose The current paper aims to develop a reduced order discontinuous Galerkin method for solving the generalized Swift–Hohenberg equation with application in biological science and mechanical engineering. The generalized Swift–Hohenberg equation is a fourth-order PDE; thus, this paper uses the local discontinuous Galerkin (LDG) method for it. Design/methodology/approach At first, the spatial direction has been discretized by the LDG technique, as this process results in a nonlinear system of equations based on the time variable. Thus, to achieve more accurate outcomes, this paper uses an exponential time differencing scheme for solving the obtained system of ordinary differential equations. Finally, to decrease the used CPU time, this study combines the proper orthogonal decomposition approach with the LDG method and obtains a reduced order LDG method. The circular and rectangular computational domains have been selected to solve the generalized Swift–Hohenberg equation. Furthermore, the energy stability for the semi-discrete LDG scheme has been discussed. Findings The results show that the new numerical procedure has not only suitable and acceptable accuracy but also less computational cost compared to the local DG without the proper orthogonal decomposition (POD) approach. Originality/value The local DG technique is an efficient numerical procedure for solving models in the fluid flow. The current paper combines the POD approach and the local LDG technique to solve the generalized Swift–Hohenberg equation with application in the fluid mechanics. In the new technique, the computational cost and the used CPU time of the local DG have been reduced.

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