scholarly journals Transition to chaos in a reduced-order model of a shear layer

2021 ◽  
Vol 932 ◽  
Author(s):  
André V.G. Cavalieri ◽  
Erico L. Rempel ◽  
Petrônio A.S. Nogueira
Keyword(s):  
Boundary Conditions ◽  
Shear Layer ◽  
Amplitude Modulation ◽  
Shear Layers ◽  
Reduced Order Model ◽  
Galerkin Projection ◽  
Chaotic Attractors ◽  
Order Model ◽  
Reduced Order ◽  
Transition To Chaos

The present work studies the nonlinear dynamics of a shear layer, driven by a body force and confined between parallel walls, a simplified setting to study transitional and turbulent shear layers. It was introduced by Nogueira & Cavalieri (J. Fluid Mech., vol. 907, 2021, A32), and is here studied using a reduced-order model based on a Galerkin projection of the Navier–Stokes system. By considering a confined shear layer with free-slip boundary conditions on the walls, periodic boundary conditions in streamwise and spanwise directions may be used, simplifying the system and enabling the use of methods of dynamical systems theory. A basis of eight modes is used in the Galerkin projection, representing the mean flow, Kelvin–Helmholtz vortices, rolls, streaks and oblique waves, structures observed in the cited work, and also present in shear layers and jets. A dynamical system is obtained, and its transition to chaos is studied. Increasing Reynolds number $Re$ leads to pitchfork and Hopf bifurcations, and the latter leads to a limit cycle with amplitude modulation of vortices, as in the direct numerical simulations by Nogueira & Cavalieri. Further increase of $Re$ leads to the appearance of a chaotic saddle, followed by the emergence of quasi-periodic and chaotic attractors. The chaotic attractors suffer a merging crisis for higher $Re$ , leading to a chaotic dynamics with amplitude modulation and phase jumps of vortices. This is reminiscent of observations of coherent structures in turbulent jets, suggesting that the model represents a dynamics consistent with features of shear layers and jets.

scholarly journals A higher-order parametric nonlinear reduced-order model for imperfect structures using Neumann expansion

2021 ◽  
Author(s):  
J. Marconi ◽  
P. Tiso ◽  
D. E. Quadrelli ◽  
F. Braghin
Keyword(s):  
Reduced Order Model ◽  
Galerkin Projection ◽  
Order Model ◽  
Displacement Fields ◽  
Reduced Order ◽  
Total Displacement ◽  
Elastic Forces ◽  
Strain Formulation ◽  
The One ◽  
Definition Of

AbstractWe present an enhanced version of the parametric nonlinear reduced-order model for shape imperfections in structural dynamics we studied in a previous work. In this model, the total displacement is split between the one due to the presence of a shape defect and the one due to the motion of the structure. This allows to expand the two fields independently using different bases. The defected geometry is described by some user-defined displacement fields which can be embedded in the strain formulation. This way, a polynomial function of both the defect field and actual displacement field provides the nonlinear internal elastic forces. The latter can be thus expressed using tensors, and owning the reduction in size of the model given by a Galerkin projection, high simulation speedups can be achieved. We show that the adopted deformation framework, exploiting Neumann expansion in the definition of the strains, leads to better accuracy as compared to the previous work. Two numerical examples of a clamped beam and a MEMS gyroscope finally demonstrate the benefits of the method in terms of speed and increased accuracy.

scholarly journals Modeling the Electrostatic Deflection of a MEMS Multilayers Based Actuator

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Hassen M. Ouakad ◽  
Muhammad A. Hawwa ◽  
Hussain M. Al-Qahtani
Keyword(s):  
Finite Element ◽  
Boundary Conditions ◽  
Integrodifferential Equations ◽  
Reduced Order Model ◽  
Rigid Substrate ◽  
Order Model ◽  
Commercial Software ◽  
Reduced Order ◽  
Electrostatic Deflection

An actuator comprised of a rigid substrate and two parallel clamped-clamped microbeams is modeled under the influence of electrostatic loading. The problem is considered under the context of nonlinear Euler's mechanics, where the actuating system is described by coupled integrodifferential equations with relevant boundary conditions. Galerkin-based discretization is utilized to obtain a reduced-order model, which is solved numerically. Actuators with different gap sizes between electrode and beams are investigated. The obtained results are compared to simulations gotten by the finite-element commercial software ANSYS.

scholarly journals Evolve Then Filter Regularization for Stochastic Reduced Order Modeling

2018 ◽  
Author(s):  
Xuping Xie ◽  
Feng Bao ◽  
Clayton G. Webster
Keyword(s):  
Stochastic Systems ◽  
Spatial Filter ◽  
Reduced Order Modeling ◽  
Reduced Order Model ◽  
Galerkin Projection ◽  
Order Model ◽  
Reduced Order ◽  
Stochastic Burgers Equation ◽  
Numerical Stabilization ◽  
Convection Dominated

In this paper, we introduce the evolve-then-filter (EF) regularization method for reduced order modeling of convection-dominated stochastic systems. The standard Galerkin projection reduced order model (G-ROM) yield numerical oscillations in a convection-dominated regime. The evolve-then-filter reduced order model (EF-ROM) aims at the numerical stabilization of the standard G-ROM, which uses explicit ROM spatial filter to regularize various terms in the reduced order model (ROM). Our numerical results based on a stochastic Burgers equation with linear multiplicative noise. It shows that the EF-ROM is significantly better results than G-ROM.

Data-driven construction of a reduced-order model for supersonic boundary layer transition

2019 ◽  
Vol 874 ◽  
pp. 1096-1114 ◽  
Author(s):  
Ming Yu ◽  
Wei-Xi Huang ◽  
Chun-Xiao Xu
Keyword(s):  
Compressible Flows ◽  
Incompressible Flows ◽  
Temporal Evolution ◽  
Reduced Order Model ◽  
Boundary Layer Transition ◽  
Galerkin Projection ◽  
Data Driven ◽  
Dynamic Mode ◽  
Order Model ◽  
Reduced Order

In this study, a data-driven method for the construction of a reduced-order model (ROM) for complex flows is proposed. The method uses the proper orthogonal decomposition (POD) modes as the orthogonal basis and the dynamic mode decomposition method to obtain linear equations for the temporal evolution coefficients of the modes. This method eliminates the need for the governing equations of the flows involved, and therefore saves the effort of deriving the projected equations and proving their consistency, convergence and stability, as required by the conventional Galerkin projection method, which has been successfully applied to incompressible flows but is hard to extend to compressible flows. Using a sparsity-promoting algorithm, the dimensionality of the ROM is further reduced to a minimum. The ROMs of the natural and bypass transitions of supersonic boundary layers at $Ma=2.25$ are constructed by the proposed data-driven method. The temporal evolution of the POD modes shows good agreement with that obtained by direct numerical simulations in both cases.

scholarly journals New approximations, and policy implications, from a delayed dynamic model of a fast pandemic

2020 ◽  
Author(s):  
C. P. Vyasarayani ◽  
Anindya Chatterjee
Keyword(s):  
Numerical Solutions ◽  
Initial Conditions ◽  
Reduced Order Model ◽  
Policy Implications ◽  
Galerkin Projection ◽  
Order Model ◽  
Reduced Order ◽  
Wave Approximation ◽  
Long Wave ◽  
Long Wave Approximation

AbstractWe study an SEIQR (Susceptible-Exposed-Infectious-Quarantined-Recovered) model for an infectious disease, with time delays for latency and an asymptomatic phase. For fast pandemics where nobody has prior immunity and everyone has immunity after recovery, the SEIQR model decouples into two nonlinear delay differential equations (DDEs) with five parameters. One parameter is set to unity by scaling time. The subcase of perfect quarantining and zero self-recovery before quarantine, with two free parameters, is examined first. The method of multiple scales yields a hyperbolic tangent solution; and a long-wave approximation yields a first order ordinary differential equation (ODE). With imperfect quarantining and nonzero self-recovery, the long-wave approximation is a second order ODE. These three approximations each capture the full outbreak, from infinitesimal initiation to final saturation. Low-dimensional dynamics in the DDEs is demonstrated using a six state non-delayed reduced order model obtained by Galerkin projection. Numerical solutions from the reduced order model match the DDE over a range of parameter choices and initial conditions. Finally, stability analysis and numerics show how correctly executed time-varying social distancing, within the present model, can cut the number of affected people by almost half. Alternatively, faster detection followed by near-certain quarantining can potentially be even more effective.

scholarly journals POD-Galerkin reduced order methods for CFD using Finite Volume Discretisation: vortex shedding around a circular cylinder

2017 ◽  
Vol 8 (1) ◽  
pp. 210-236 ◽  
Author(s):  
Giovanni Stabile ◽  
Saddam Hijazi ◽  
Andrea Mola ◽  
Stefano Lorenzi ◽  
Gianluigi Rozza
Keyword(s):  
Circular Cylinder ◽  
Finite Volume ◽  
Vortex Shedding ◽  
Dimensional Space ◽  
Reduced Order Model ◽  
Galerkin Projection ◽  
Governing Equations ◽  
Reduced Basis ◽  
Order Model ◽  
Reduced Order

Abstract Vortex shedding around circular cylinders is a well known and studied phenomenon that appears in many engineering fields. A Reduced Order Model (ROM) of the incompressible ow around a circular cylinder is presented in this work. The ROM is built performing a Galerkin projection of the governing equations onto a lower dimensional space. The reduced basis space is generated using a Proper Orthogonal Decomposition (POD) approach. In particular the focus is into (i) the correct reproduction of the pres- sure field, that in case of the vortex shedding phenomenon, is of primary importance for the calculation of the drag and lift coefficients; (ii) the projection of the Governing equations (momentum equation and Poisson equation for pressure) performed onto different reduced basis space for velocity and pressure, respectively; (iii) all the relevant modifications necessary to adapt standard finite element POD-Galerkin methods to a finite volume framework. The accuracy of the reduced order model is assessed against full order results.

scholarly journals Evolve Filter Stabilization Reduced-Order Model for Stochastic Burgers Equation

Fluids ◽  
2018 ◽  
Vol 3 (4) ◽  
pp. 84
Author(s):  
Xuping Xie ◽  
Feng Bao ◽  
Clayton Webster
Keyword(s):  
Stochastic Systems ◽  
Burgers Equation ◽  
Spatial Filter ◽  
Reduced Order Model ◽  
Galerkin Projection ◽  
Order Model ◽  
Reduced Order ◽  
Numerical Result ◽  
Stochastic Burgers Equation ◽  
Convection Dominated

In this paper, we introduce the evolve-then-filter (EF) regularization method for reduced order modeling of convection-dominated stochastic systems. The standard Galerkin projection reduced order model (G-ROM) yield numerical oscillations in a convection-dominated regime. The evolve-then-filter reduced order model (EF-ROM) aims at the numerical stabilization of the standard G-ROM, which uses explicit ROM spatial filter to regularize various terms in the reduced order model (ROM). Our numerical results are based on a stochastic Burgers equation with linear multiplicative noise. The numerical result shows that the EF-ROM is significantly better than G-ROM.

Reduced-Order Model for Unsteady Compressible Flow Based on POD-Galerkin Projection

2012 ◽  
Vol 226-228 ◽  
pp. 835-839
Author(s):  
Yi Bin Dou ◽  
Min Xu
Keyword(s):  
Euler Equation ◽  
Compressible Flow ◽  
Calibration Method ◽  
Initial Guess ◽  
Reduced Order Model ◽  
Galerkin Projection ◽  
Numerical Dissipation ◽  
Test Case ◽  
Order Model ◽  
Reduced Order

Italic textIn this paper, the reduced-order model (ROM) of unsteady compressible flow based on POD-Galerkin projection has been investigated. The Euler equation formulated with the conservative variables for compressible flow has been reformulated using modified primitive variables. The POD modes are computed using snapshot method and then an explicit quadratic ROM is constructed by applying the Galerkin projection to the modified Euler equation. Because of lacking any dissipation in POD-Galerkin projection, the flow calibration method is introduced to account for the numerical dissipation to stabilize the ROM. At last, the NACA0012 airfoil undergoing pitch harmonic oscillating in Mach number Mach = 0.755 is calculated as a test case. For this flow configuration, the calibrated coefficients of the ROM are almost the same as the initial guess which comes from POD-Galerkin projection; the differences are mostly focused on the linear Lij coefficients and quadratic Qijk coefficients.

Closure modeling in reduced-order model of Burgers’ equation for control applications

2016 ◽  
Vol 231 (4) ◽  
pp. 642-656 ◽  
Author(s):  
Haroon Imtiaz ◽  
Imran Akhtar
Keyword(s):  
Boundary Conditions ◽  
Turbulent Flows ◽  
Reduced Order Model ◽  
Reduced Order Models ◽  
Closure Model ◽  
Order Model ◽  
Control Applications ◽  
Reduced Order ◽  
Nonhomogeneous Boundary ◽  
Nonhomogeneous Boundary Conditions

A proper orthogonal decomposition (POD) technique has been successfully employed to develop reduced-order models for flow control purposes. For complex flows, higher POD modes also play a significant role in the stability and accuracy of the reduced-order model, thus require a closure, as in turbulent flows. In the presence of nonhomogeneous boundary conditions, developing a closure model becomes a challenging task. This paper discusses nonlinear closure modeling approaches for homogeneous and nonhomogeneous boundary conditions. Burgers’ equations, both one-dimensional and two-dimensional, are considered as the governing equations to develop reduced-order models with different boundary conditions. Homogeneous and nonhomogeneous boundary conditions are considered to demonstrate the effectiveness of the proposed closure modeling technique in boundary control applications. Numerical results show that the proposed closure model improves the accuracy of the reduced-order model.

Calibration of reduced-order model for a coupled Burgers equations based on PC-EnKF

2018 ◽  
Vol 13 (2) ◽  
pp. 21 ◽  
Author(s):  
Yuepeng Wang ◽  
Yue Cheng ◽  
Zongyuan Zhang ◽  
Guang Lin
Keyword(s):  
Kalman Filter ◽  
Ensemble Kalman Filter ◽  
Interpolation Method ◽  
Initial Guess ◽  
Reduced Order Model ◽  
Galerkin Projection ◽  
Order Model ◽  
Inference Problem ◽  
Reduced Order ◽  
Burgers Equations

The proper orthogonal decomposition (POD) and the discrete empirical interpolation method (DEIM) are applied to coupled Burgers equations to develop its reduced-order model (ROM) by the Galerkin projection. A calibrated POD ROM is developed in the current study through adding and multiplying a set of time-dependent random parameters to recover the loss of accuracy due to the truncation of the POD modes. Calibrating the ROM becomes essentially a high-dimensional statistical inverse inference problem. To reduce the computational effort, the polynomial chaos based ensemble Kalman filter (PC-EnKF) is adopted in this work. By using a sparse optimization algorithm, a sparse PC expansion is obtained to facilitate further calculation of statistical moments used in ensemble Kalman filter. We apply the well-defined calibrated POD ROM for the coupled Burgers equations with the Reynolds numberRe= 10 000. The numerical results show that the PC-EnKF method is efficient in reducing the uncertainty included in the initial guess of input parameters and feasible in correcting the behavior of the POD ROM. The study suggests that the PC-EnKF is quite general as a calibration tool for calibrating the POD ROM.

Sign in / Sign up

Export Citation Format

Share Document