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1

Stabilized model reduction for nonlinear dynamical systems through a contractivity-preserving framework

Chaturantabut Saifon

International Journal of Applied Mathematics and Computer Science

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2020

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Vol. 30, no. 4

615--628

EN

This work develops a technique for constructing a reduced-order system that not only has low computational complexity, but also maintains the stability of the original nonlinear dynamical system. The proposed framework is designed to preserve the contractivity of the vector field in the original system, which can further guarantee stability preservation, as well as provide an error bound for the approximated equilibrium solution of the resulting reduced system. This technique employs a low-dimensional basis from proper orthogonal decomposition to optimally capture the dominant dynamics of the original system, and modifies the discrete empirical interpolation method by enforcing certain structure for the nonlinear approximation. The efficiency and accuracy of the proposed method are illustrated through numerical tests on a nonlinear reaction diffusion problem.

2

On the extrapolated VMS-POD method for incompressible flows

Eroglu Fatma G., Kaya Songul

Computer Methods in Materials Science

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2019

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Vol. 19, No. 2

70--77

EN

In this study, proper orthogonal decomposition based reduced-order modelling and variational multiscale stabilization method are utilized for the incompressible Navier-Stokes equations. In addition, the difficulties resulting from nonlinearity are eliminated using the extrapolation. Theoretical analysis of the method is presented. To check the efficiency of the proposed method, we utilize the test problem of 2D channel flow past a cylinder.

PL

W artykule zastosowano obniżenie rzędu modelu i wariacyjną wieloskalową stabilizacię, wykorzystujące ortogonalną dekompozycję, do rozwiązania równania Naviera-Stokesa dla nieściśliwych płynów. Trudności wynikające z nieliniowości wyeliminowano poprzez zastosowanie ekstrapolacji. W pracy opisano teoretyczne podstawy metody. W celu sprawdzenia wydajności opracowanej metody zastosowano ję do testowego problemu przepływu 2D przez cylindryczny kanał.

3

On streamwise structures in boundary layer under adverse pressure gradient on inclined plate

Uruba V., Prochazka P., Skala V.

TASK Quarterly : scientific bulletin of Academic Computer Centre in Gdansk

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2018

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Vol. 22, No 3

255--267

EN

The presented study is focused on experimental investigation of a boundary layer on a flat plate in an adverse pressure gradient. The flat plate is placed in a regular flow, the pressure gradient is generated by the plate inclination. The preceding studies deal with the structure of the wake behind the plate, the presented study concentrates on the flow structureclose to the suction surface of the plate. The dynamical behavior of the flow structures is studied in details with respect to the topology in the streamwise direction. In spite of the fact that thetime-mean flow field is close to 2D, more or less constant along the span, the instantaneous structures topology is fully 3D. Rather oblique structures are detected instead of those oriented in the streamwise direction. The patterns are travelling in the streamwise direction along the plate.

4

Bizon K.

Chemical and Process Engineering

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2015

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Vol. 36, nr 3

365--376

EN

Over the last decades the method of proper orthogonal decomposition (POD) has been successfully employed for reduced order modelling (ROM) in many applications, including distributed parameter models of chemical reactors. Nevertheless, there are still a number of issues that need further investigation. Among them, the policy of the collection of representative ensemble of experimental or simulation data, being a starting and perhaps most crucial point of the POD-based model reduction procedure. This paper summarises the theoretical background of the POD method and briefly discusses the sampling issue. Next, the reduction procedure is applied to an idealised model of circulating fluidised bed combustor (CFBC). Results obtained confirm that a proper choice of the sampling strategy is essential for the modes convergence however, even low number of observations can be sufficient for the determination of the faithful dynamical ROM.

5

Optimality system POD and a-posteriori error analysis for linear-quadratic problems

Volkwein S.

Control and Cybernetics

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2011

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Vol. 40, no 4

1109-1124

EN

In this paper an abstract linear-quadratic optimal control problem governed by an evolution equation is considered. To solve this problem numerically a reduced-order approach based on proper orthogonal decomposition (POD) is applied. The error between the POD suboptimal control and the optimal control of the original problem is controlled by an a-posteriori error analysis. However, if the POD basis has bad approximation properties, a huge number of POD basis function is required to solve the reduced-order problem with the desired accuracy. To overcome this problem, optimality system POD (OS-POD) is utilized, where the POD basis is chosen with respect to the optimization criteria.

6

Reduced order models in PIDE constrained optimization

Sachs E. W., Schu M.

Control and Cybernetics

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2010

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Vol. 39, no 3

661-675

EN

Mathematical models for option pricing often result in partial differential equations originally starting with the Black-Scholes model. In this context, recent enhancements are models driven by Levy processes, which lead to a partial differential equation with an additional integral term. If one solves the problems mentioned last numerically, this yields large linear systems of equations with dense matrices. However, by using the special structure and an iterative solver the problem can be handled efficiently. To further reduce the computational cost in the calibration phase we implement a reduced order model, like proper orthogonal decomposition (POD), which proves to be very efficient. In this paper we use a special multi-level trust region POD algorithm to calibrate the option pricing model and give numerical results supporting the gain in efficiency.

7

A 3D CFD model of a natural draft wet-cooling tower

Klimanek A., Białecki R.

Archives of Thermodynamics

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2009

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Vol. 30, no. 4

119-132

EN

A 3D CFD model of a natural draft wet-cooling tower is presented in this paper. The model encompasses both the interior of the tower as well as the surrounding air. The developed CFD model is supplied with a low dimensional representation of the heat and mass exchanger, whose purpose is to determine the heat and mass rejected from the cooled water to the air. The representation is based on an original technique called the proper orthogonal decomposition. Due to the large scale differences, application of a low-dimensional heat and mass transfer model allowed reducing the computational time and accurately predict the heat and mass rejection effects. The Euler-Euler multiphase model was used to calculate the flow, heat and mass transfer in the rain zone. The model can be used in both, design computations as well as performance tests of natural draft wet-cooling towers. The effects of wind on the cooling tower can be taken into account. The CFD model was developed using the commercial code Fluent.

8

Modelling of near-wall turbulence with large-eddy velocity modes

Wacławczyk M., Pozorski J.

Journal of Theoretical and Applied Mechanics

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2007

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Vol. 45 nr 3

705-724

EN

In the paper, low-order modelling of the turbulent velocity field in the near-wall region is performed using the Proper Orthogonal Decomposition (POD) approach. First, an empirical eigenfunction basis is computed, basing on two-point velocity correlations. Next, the Galerkin projection of the Navier-Stokes equations on the truncated basis is performed. This results in a system of Ordinary Differential Equations (ODEs) for time-dependent coefficients. Evolution of the largest vortical structures in the near-wall zone is then obtained from the time dependent coefficients and eigenfunctions. The system applied in the present work consists of 20 ODEs, the reconstructed velocity field is two-dimensional in the pIane perpendicular to the main flow direction. Moreover, the filtering procedure associated with the POD method is discussed, the POD filter is derived and compared with LES filters.

PL

Przedmiotem pracy jest modelowanie turbulentnego pola prędkości w obszarze przyściennym za pomocą niskowymiarowego systemu dynamicznego, opartego o dekompozycję w bazie funkcji własnych POD (ang. Proper Orthogonal Decomposition). Empiryczna baza funkcyjna POD została wyznaczona z rozwiązania zagadnienia własnego, w którym obecne są dwupunktowe korelacje prędkości. Następnie, w wyniku projekcji Galerkina równań pędu na podprzestrzeń rozpiętą na tej bazie funkcyjnej, otrzymano układ równań różniczkowych zwyczajnych na zależne od czasu wspołczynniki. Na podstawie funkcji własnych oraz z wyznaczonych współczynników rozkładu uzyskano ewolucję w czasie charakterystycznych struktur wirowych w obszarze przyściennym. System dynamiczny rozpatrywany w pracy składa się z 20 równań różniczkowych zwyczajnych. Zrekonstruowane pole prędkości jest dwuwymiarowe (w płaszczyźnie prostopadłej do głównego kierunku przepływu). Ponadto w pracy dyskutowana jest procedura filtrowania związana z metodą POD. Wyprowadzony filtr POD porównano z formułą używaną w metodzie symulacji dużych wirów.

9

Monitoring of Railway Vehicles : Detection of Critical Rolling States of Railway Wheelsets

Glösmann P., Kreuzer E.

Machine Dynamics Problems

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2005

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Vol. 29, No. 4

47-58

EN

This paper discusses the Karhunen-Loeve Transform as a methode to investigate the behavior of highly nonlinear dynamical systems. First, the mathematical concept of the Karhunen-Loeve Transform is introduced in brief. Then, the results of the joint research project "Monitoring of Railway Vehicles" are discussed. Finally, recent analyzes of the dynamics of railway wheelset by means of Karhunen-Loeve Transform are presented.

10

Rekonstrukcja geometrii obiektów przestrzennych przy zredukowanej ilości danych pomiarowych

Rychlik M.

Eksploatacja i Niezawodność

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2004

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nr 2

45-49

PL

Budowa trójwymiarowych modeli używanych w systemach CAx, wymaga precyzyjnych danych geometrycznych. W zależności od aplikacji, konieczne jest pozyskanie (oprócz danych geometrycznych) dodatkowych danych takich jak: gęstość, tekstura powierzchni, itp. Autor prezentuje metody rekonstrukcji trójwymiarowych modeli w programach CAD i przedstawia koncepcją praktycznego zastosowania POD (Proper Orthogonal Decomposition). Trójwymiarowa rekonstrukcja POD bazuje na modach statystycznych opisujących obiekt 3D.

EN

The construction of three-dimensional models used in CAx systems, to require precision geometrical data of the real object is presented. Depending on application, it is necessary to obtain (besides geometrical dimensions) additional data such as: volume density, texture of the surface, etc. Author shows method of reconstruction 3D models in CAD software and presents conception of practicable Proper Orthogonal Decomposition (POD). The POD 3D reconstruction bases on the statistical modes which describe the 3D object.

11

Reduced order controllers for Burgers' equation with a nonlinear observer

Atwell J. A., Borggaard J. T., King B. B.

International Journal of Applied Mathematics and Computer Science

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2001

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Vol. 11, no 6

1311-1330

EN

A method for reducing controllers for systems described by partial differential equations (PDEs) is applied to Burgers' equation with periodic boundary conditions. This approach differs from the typical approach of reducing the model and then designing the controller, and has developed over the past several years into its current form. In earlier work it was shown that functional gains for the feedback control law served well as a dataset for reduced order basis generation via the proper orthogonal decomposition (POD). However, the test problem was the two-dimensional heat equation, a problem in which the physics dominates the system in such a way that controller efficacy is difficult to generalize. Here, we additionally incorporate a nonlinear observer by including the nonlinear terms of the state equation in the differential equation for the compensator.