A Cantilever Conveying Fluid: Coherent Modes Versus Beam Modes

2002 ◽  
Author(s):  
A. Sarkar ◽  
M. P. Paidoussis
Keyword(s):  
Eigenvalue Problem ◽  
Cantilever Beam ◽  
Large Scale ◽  
Galerkin Projection ◽  
Reduced Order ◽  
Non Linear ◽  
Proper Orthogonal Modes ◽  
Linear Oscillation ◽  
Proper Orthogonal ◽  
Conveying Fluid

A semianalytical approach to obtain the proper orthogonal modes (POMs) is described for the non-linear oscillation of a cantilevered pipe conveying fluid. Theoretically, while the spatial coherent structures are the eigenfunctions of the time-averaged spatial autocorrelation functions, it emerges that once the Galerkin projection of the proper orthogonal modes is carried out using the uniform cantilever-beam modes, the spatial dependency of the integral eigenvalue problem can be eliminated by analytical manipulation which avoids any spatial discretization error. As the solution of the integral equation is obtained semianalytically by linearly projecting the proper orthogonal modes on the cantilever-beam modes, any linear or non-linear operation can be carried out analytically on the proper orthogonal modes. Futhermore, the reduced-order eigenvalue problem minimizes the numerical pollution which leads to spurious eigenvectors, as may arise in the case of a large-scale eigenvalue problem (without the Galerkin projection of the eigenvectors on the cantilever-beam modes). This methodology can conveniently be used to study the convergence of the numerically calculated proper or-thogonal modes obtained from the full-scale eigenvalue problem.

scholarly journals GENETIC ALGORITHM-BASED CALIBRATION OF REDUCED ORDER GALERKIN MODELS

2011 ◽  
Vol 16 (1) ◽  
pp. 233-247 ◽  
Author(s):  
Witold Stankiewicz ◽  
Robert Roszaka ◽  
Marek Morzyńskia
Keyword(s):  
Genetic Algorithm ◽  
Bluff Body ◽  
Calibration Procedure ◽  
Galerkin Projection ◽  
Reduced Order Models ◽  
Data Set ◽  
Reduced Order ◽  
Low Dimensional ◽  
Fluid Behaviour ◽  
Proper Orthogonal

Low-dimensional models, allowing quick prediction of fluid behaviour, are key enablers of closed-loop flow control. Reduction of the model's dimension and inconsistency of high-fidelity data set and the reduced-order formulation lead to the decrease of accuracy. The quality of Reduced-Order Models might be improved by a calibration procedure. It leads to global optimization problem which consist in minimizing objective function like the prediction error of the model. In this paper, Reduced-Order Models of an incompressible flow around a bluff body are constructed, basing on Galerkin Projection of governing equations onto a space spanned by the most dominant eigenmodes of the Proper Orthogonal Decomposition (POD). Calibration of such models is done by adding to Galerkin System some linear and quadratic terms, which coefficients are estimated using Genetic Algorithm.

scholarly journals POD-Galerkin Model for Incompressible Single-Phase Flow in Porous Media

Open Physics ◽  
2016 ◽  
Vol 14 (1) ◽  
pp. 588-601 ◽  
Author(s):  
Yi Wang ◽  
Bo Yu ◽  
Shuyu Sun
Keyword(s):  
Porous Media ◽  
Large Scale ◽  
Single Phase ◽  
Galerkin Projection ◽  
Flow In Porous Media ◽  
Computational Time ◽  
Speed Up ◽  
Fast Prediction ◽  
Proper Orthogonal ◽  
Phase Fluid

AbstractFast prediction modeling via proper orthogonal decomposition method combined with Galerkin projection is applied to incompressible single-phase fluid flow in porous media. Cases for different configurations of porous media, boundary conditions and problem scales are designed to examine the fidelity and robustness of the model. High precision (relative deviation 1.0 × 10−4% ~ 2.3 × 10−1%) and large acceleration (speed-up 880 ~ 98454 times) of POD model are found in these cases. Moreover, the computational time of POD model is quite insensitive to the complexity of problems. These results indicate POD model is especially suitable for large-scale complex problems in engineering.

Performance of Reduced Order Models of Moving Heat Source Deposition Problems for Efficient Inverse Analysis

2014 ◽  
Author(s):  
John G. Michopoulos ◽  
Brian Dennis ◽  
Foteini Komninelli ◽  
Athanasios Iliopoulos ◽  
Ashkan Akbariyeh
Keyword(s):  
Heat Source ◽  
Inverse Analysis ◽  
Galerkin Projection ◽  
Reduced Order Models ◽  
Moving Heat Source ◽  
Reduced Order ◽  
Computational Implementation ◽  
Model Size ◽  
Transfer Problems ◽  
Proper Orthogonal

In order to reduce the demanding computational requirements for the numerical solution of problems involving heat transfer problems of moving heat source deposition, we present an approach utilizing reduced order models based on proper orthogonal decomposition and associated Galerkin projection. We subsequently describe the finite element implementation of solution methodology for both the full order and the reduced order models, as well as the respective computational implementation details. Using this methodology, we performed a sensitivity analysis for a problem of a moving heat source to investigate the performance characteristics of the relevant reduced order model size and present the efficiency of the approach. We demonstrated the efficiency of the reduced models for performing inverse analysis.

Nonlinear Modeling of Mechanical Gas Face Seal Systems Using Proper Orthogonal Decomposition

2006 ◽  
Vol 128 (4) ◽  
pp. 817-827 ◽  
Author(s):  
Haojiong Zhang ◽  
Brad A. Miller ◽  
Robert G. Landers
Keyword(s):  
Proper Orthogonal Decomposition ◽  
Initial Conditions ◽  
Nonlinear Modeling ◽  
Orthogonal Decomposition ◽  
Galerkin Projection ◽  
Face Seal ◽  
Reduced Order ◽  
Wide Range ◽  
Gas Face Seal ◽  
Proper Orthogonal

An approach based on proper orthogonal decomposition and Galerkin projection is presented for developing low-order nonlinear models of the gas film pressure within mechanical gas face seals. A technique is developed for determining an optimal set of global basis functions for the pressure field using data measured experimentally or obtained numerically from simulations of the seal motion. The reduced-order gas film models are shown to be computationally efficient compared to full-order models developed using the conventional semidiscretization methods. An example of a coned mechanical gas face seal in a flexibly mounted stator configuration is presented. Axial and tilt modes of stator motion are modeled, and simulation studies are conducted using different initial conditions and force inputs. The reduced-order models are shown to be applicable to seals operating within a wide range of compressibility numbers, and results are provided that demonstrate the global reduced-order model is capable of predicting the nonlinear gas film forces even with large deviations from the equilibrium clearance.

scholarly journals Computation of lossy higher order modes in complex SRF cavities using Beyn’s and Newton’s methods on reduced order models

2019 ◽  
Vol 34 (36) ◽  
pp. 1942037
Author(s):  
Hermann W. Pommerenke ◽  
Johann D. Heller ◽  
Shahnam Gorgi Zadeh ◽  
Ursula van Rienen
Keyword(s):  
Eigenvalue Problem ◽  
Large Scale ◽  
Nonlinear Eigenvalue Problem ◽  
Real Life ◽  
Light Sources ◽  
Reduced Order ◽  
Performance Requirements ◽  
Precise Knowledge ◽  
Superconducting Cavities ◽  
Electromagnetic Resonances

Superconducting radio frequency cavities meet the demanding performance requirements of modern accelerators and high-brilliance light sources. Their design requires a precise knowledge of their electromagnetic resonances. A numerical solution of Maxwell’s equations is required to compute the resonant eigenmodes, their frequencies and losses due to the complex cavity shape. The consideration of resonances damped by external losses leads to a nonlinear eigenvalue problem. Previous work showed that, using State-Space Concatenation to construct a reduced order model and Newton iteration to solve the arising eigenvalue problem, solutions can be obtained on workstation computers even for large-scale problems without extensive simplification of the structure itself. In this paper, we augment the solution workflow by Beyn’s contour integral algorithm to increase the number of found eigenmodes. Numerical experiments are presented for one academic and two real-life superconducting cavities and partially compared to measurements.

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.

Error Estimates for Reduced Order POD Models of Navier-Stokes Equations

2008 ◽  
Author(s):  
S. S. Ravindran
Keyword(s):  
Proper Orthogonal Decomposition ◽  
Error Estimates ◽  
Incompressible Flows ◽  
Orthogonal Decomposition ◽  
Galerkin Projection ◽  
Dimensional Model ◽  
Reduced Order ◽  
Infinite Dimensional ◽  
Low Dimensional ◽  
Proper Orthogonal

Reduced order modeling for the purpose of constructing a low dimensional model from high dimensional or infinite dimensional model has important applications in science and engineering such as fast model evaluations and optimization/control. A popular method for constructing reduced-order model is based on finding a suitable low dimensional basis by proper orthogonal decomposition (POD) and forming a model by Galerkin projection of the infinite dimensional model onto the basis. In this paper, we will discuss error estimates for Galerkin proper orthogonal decomposition method for an unsteady nonlinear coupled partial differential equations arising in viscous incompressible flows. A specific finite element in space and finite difference in time discretization scheme will be discussed.

Equation-Free/Galerkin-Free Reduced-Order Modeling of the Shallow Water Equations Based on Proper Orthogonal Decomposition

2009 ◽  
Vol 131 (7) ◽  
Author(s):  
Vahid Esfahanian ◽  
Khosro Ashrafi
Keyword(s):  
Shallow Water ◽  
Basis Functions ◽  
Galerkin Projection ◽  
Scale Model ◽  
Fine Scale ◽  
Coarse Scale ◽  
Reduced Order ◽  
Shallow Water Flows ◽  
The One ◽  
Proper Orthogonal

In this paper, two categories of reduced-order modeling (ROM) of the shallow water equations (SWEs) based on the proper orthogonal decomposition (POD) are presented. First, the traditional Galerkin-projection POD/ROM is applied to the one-dimensional (1D) SWEs. The result indicates that although the Galerkin-projection POD/ROM is suitable for describing the physical properties of flows (during the POD basis functions’ construction time), it cannot predict that the dynamics of the shallow water flows properly as it was expected, especially with complex initial conditions. Then, the study is extended to applying the equation-free/Galerkin-free POD/ROM to both 1D and 2D SWEs. In the equation-free/Galerkin-free framework, the numerical simulation switches between a fine-scale model, which provides data for construction of the POD basis functions, and a coarse-scale model, which is designed for the coarse-grained computational study of complex, multiscale problems like SWEs. In the present work, the Beam & Warming and semi-implicit time integration schemes are applied to the 1D and 2D SWEs, respectively, as fine-scale models and the coefficients of a few POD basis functions (reduced-order model) are considered as a coarse-scale model. Projective integration is applied to the coarse-scale model in an equation-free framework with a time step grater than the one used for a fine-scale model. It is demonstrated that equation-free/Galerkin-free POD/ROM can resolve the dynamics of the complex shallow water flows. Moreover, the computational cost of the approach is less than the one for a fine-scale model.

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