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.

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.

Modeling the Turbulent Wake Behind a Wall-Mounted Square Cylinder

2020 ◽  
Author(s):  
Christian Amor ◽  
José M Pérez ◽  
Philipp Schlatter ◽  
Ricardo Vinuesa ◽  
Soledad Le Clainche
Keyword(s):  
Proper Orthogonal Decomposition ◽  
Turbulent Wake ◽  
Orthogonal Decomposition ◽  
Dynamic Mode Decomposition ◽  
Dynamic Mode ◽  
Square Cylinder ◽  
Complex Flow ◽  
Reduced Order ◽  
Mode Decomposition ◽  
Proper Orthogonal

Abstract This article introduces some soft computing methods generally used for data analysis and flow pattern detection in fluid dynamics. These techniques decompose the original flow field as an expansion of modes, which can be either orthogonal in time (variants of dynamic mode decomposition), or in space (variants of proper orthogonal decomposition) or in time and space (spectral proper orthogonal decomposition), or they can simply be selected using some sophisticated statistical techniques (empirical mode decomposition). The performance of these methods is tested in the turbulent wake of a wall-mounted square cylinder. This highly complex flow is suitable to show the ability of the aforementioned methods to reduce the degrees of freedom of the original data by only retaining the large scales in the flow. The main result is a reduced-order model of the original flow case, based on a low number of modes. A deep discussion is carried out about how to choose the most computationally efficient method to obtain suitable reduced-order models of the flow. The techniques introduced in this article are data-driven methods that could be applied to model any type of non-linear dynamical system, including numerical and experimental databases.

Reduced-Order Model of a Bladed Rotor With Geometric Mistuning

2007 ◽  
Author(s):  
Alok Sinha
Keyword(s):  
Proper Orthogonal Decomposition ◽  
Orthogonal Decomposition ◽  
Reduced Order Model ◽  
Order Model ◽  
Reduced Order ◽  
Coordinate Measurement ◽  
Coordinate Measurement Machine ◽  
Bladed Disk ◽  
Bladed Rotor ◽  
Proper Orthogonal

This paper deals with the development of an accurate reduced-order model of a bladed disk with geometric mistuning. The method is based on vibratory modes of various tuned systems and proper orthogonal decomposition of coordinate measurement machine (CMM) data on blade geometries. Results for an academic rotor are presented to establish the validity of the technique.

scholarly journals Local improvements to reduced-order models using sensitivity analysis of the proper orthogonal decomposition

2009 ◽  
Vol 629 ◽  
pp. 41-72 ◽  
Author(s):  
ALEXANDER HAY ◽  
JEFFREY T. BORGGAARD ◽  
DOMINIQUE PELLETIER
Keyword(s):  
Sensitivity Analysis ◽  
Proper Orthogonal Decomposition ◽  
Stokes Equations ◽  
Orthogonal Decomposition ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Reduced Order ◽  
Selection Step ◽  
Low Dimensional ◽  
Proper Orthogonal

The proper orthogonal decomposition (POD) is the prevailing method for basis generation in the model reduction of fluids. A serious limitation of this method, however, is that it is empirical. In other words, this basis accurately represents the flow data used to generate it, but may not be accurate when applied ‘off-design’. Thus, the reduced-order model may lose accuracy for flow parameters (e.g. Reynolds number, initial or boundary conditions and forcing parameters) different from those used to generate the POD basis and generally does. This paper investigates the use of sensitivity analysis in the basis selection step to partially address this limitation. We examine two strategies that use the sensitivity of the POD modes with respect to the problem parameters. Numerical experiments performed on the flow past a square cylinder over a range of Reynolds numbers demonstrate the effectiveness of these strategies. The newly derived bases allow for a more accurate representation of the flows when exploring the parameter space. Expanding the POD basis built at one state with its sensitivity leads to low-dimensional dynamical systems having attractors that approximate fairly well the attractor of the full-order Navier–Stokes equations for large parameter changes.

Patterns and Reduced-order Reconstruction of Impinging Wiping Jet Pressure Profile Fluctuation Using Proper Orthogonal Decomposition

2021 ◽  
Author(s):  
Le Quang Phan ◽  
Andrew Johnstone ◽  
P. Buyung Kosasih ◽  
Wayne Renshaw
Keyword(s):  
Proper Orthogonal Decomposition ◽  
Jet Impingement ◽  
Pressure Profile ◽  
Orthogonal Decomposition ◽  
Maximum Pressure ◽  
Frequency Spectra ◽  
Reduced Order ◽  
Order Reconstruction ◽  
Pressure Profiles ◽  
Proper Orthogonal

Abstract Wiping jet impingement pressure is important in controlling the coating mass (thickness) and influencing the smoothness of the thin metallic coating produced in continuous galvanizing lines (CGLs). However, the fluctuation of the impingement pressure profile that directly impacts the coating smoothness has not been adequately understood. To study key features of the impingement pressure fluctuation, the instantaneous impingement pressure profiles obtained from Large Eddy Simulations were analyzed using Proper Orthogonal Decomposition (POD). Dominant fluctuation modes of pressure profiles can be differentiated from the energy contents of the modes corresponding to different jet types namely mixing, non-mixing, and transitional mixing jet. The dominant modes of mixing jets in the wiping region contain comparable strength of all modes (flapping, pulsing, and out-of-phase multi pulsing). Non-mixing jets do not show discernable fluctuation modes and transitional mixing jets show pulsing and flapping modes only. Additionally, instantaneous maximum pressure gradient and their location were determined from the reduced-order reconstruction of the pressure profiles. From the analysis, frequency spectra of the magnitude and location fluctuations of the maximum pressure gradients associated with each of the jet types can be clearly distinguished. This is a knowledge that may be helpful for CGL operators in the operation of wiping jets.

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