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.

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.

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.

scholarly journals Reduced Order Models for Nonlinear Dynamic Analysis With Application to a Fan Blade

2019 ◽  
Author(s):  
Mikel Balmaseda ◽  
G. Jacquet-Richardet ◽  
A. Placzek ◽  
D.-M. Tran
Keyword(s):  
Normal Modes ◽  
Nonlinear Dynamic Analysis ◽  
Evaluation Procedure ◽  
Reduced Order Models ◽  
Reduced Basis ◽  
Reduced Order ◽  
Stiffness Evaluation ◽  
Fan Blade ◽  
Proper Orthogonal ◽  
Good Agreement

Abstract In the present work reduced order models (ROM) that are independent from the full order finite element models (FOM) considering geometrical non linearities are developed and applied to the dynamic study of a fan. The structure is considered to present nonlinear vibrations around the pre-stressed equilibrium induced by rotation enhancing the classical linearised approach. The reduced nonlinear forces are represented by a polynomial expansion obtained by the Stiffness Evaluation Procedure (STEP) and then corrected by means of a Proper Orthogonal Decomposition (POD) that filters the full order nonlinear forces (StepC ROM). The Linear Normal Modes (LNM) and Craig-Bampton (C-B) type reduced basis are considered here. The latter are parametrised with respect to the rotating velocity. The periodic solutions obtained with the StepC ROM are in good agreement with the solutions of the FOM and are more accurate than the linearised ROM solutions and the STEP ROM. The proposed StepC ROM provides the best compromise between accuracy and time consumption of the ROM.

Energy harvesting from aeroelastic vibrations induced by discrete gust loads

2016 ◽  
Vol 28 (1) ◽  
pp. 47-62 ◽  
Author(s):  
Claudia Bruni ◽  
James Gibert ◽  
Giacomo Frulla ◽  
Enrico Cestino ◽  
Pier Marzocca
Keyword(s):  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Flow Region ◽  
Rotational Displacement ◽  
Reduced Order ◽  
Piezoelectric Elements ◽  
Out Of Plane ◽  
Gust Loads ◽  
Three Degree Of Freedom ◽  
Euler Bernoulli Beam

This article evaluates the amount of energy that can be extracted from a gust using an aeroelastic energy harvester composed of a flexible wing with attached piezoelectric elements. The harvester operates in a subcritical flow region. It is modeled as a linear Euler–Bernoulli beam sandwiched between two piezoceramics. The extended Hamilton’s principle is used to derive the harvester’s equations of motion and an eigenfunction expansion is used to form a three-degree-of-freedom reduced-order model. The degrees of freedom retained in the model are two flexural degrees for the in-plane and out-of-plane displacements, and a torsional degree for the rotational displacement. Wagner and Küssner functions are used to represent the unsteady aerodynamic and gust loading, respectively. The amount of energy extracted from the system is then compared for two different deterministic gust profiles, 1-COSINE and two sharp-edged gusts forming a square gust, for various magnitudes and durations. The results show that the harvester is able to extract more energy from the square gust profile, although for both profiles the harvester extracts more power after the gust has subsided.

Sign in / Sign up

Export Citation Format

Share Document