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.

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.

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.

Reduced Order Modeling of Entrained Flow Solid Fuel Gasification

2009 ◽  
Author(s):  
Rory F. D. Monaghan ◽  
Mayank Kumar ◽  
Simcha L. Singer ◽  
Cheng Zhang ◽  
Ahmed F. Ghoniem
Keyword(s):  
Fluid Dynamics ◽  
Reduced Order Modeling ◽  
Combined Cycle ◽  
Reduced Order Model ◽  
Temperature Profiles ◽  
Order Model ◽  
External Temperature ◽  
Reduced Order ◽  
Entrained Flow ◽  
Entrained Flow Gasifiers

Reduced order models that accurately predict the operation of entrained flow gasifiers as components within integrated gasification combined cycle (IGCC) or polygeneration plants are essential for greater commercialization of gasification-based energy systems. A reduced order model, implemented in Aspen Custom Modeler, for entrained flow gasifiers that incorporates mixing and recirculation, rigorously calculated char properties, drying and devolatilization, chemical kinetics, simplified fluid dynamics, heat transfer, slag behavior and syngas cooling is presented. The model structure and submodels are described. Results are presented for the steady-state simulation of a two-metric-tonne-per-day (2 tpd) laboratory-scale Mitsubishi Heavy Industries (MHI) gasifier, fed by two different types of coal. Improvements over the state-of-the-art for reduced order modeling include the ability to incorporate realistic flow conditions and hence predict the gasifier internal and external temperature profiles, the ability to easily interface the model with plant-wide flowsheet models, and the flexibility to apply the same model to a variety of entrained flow gasifier designs. Model validation shows satisfactory agreement with measured values and computational fluid dynamics (CFD) results for syngas temperature profiles, syngas composition, carbon conversion, char flow rate, syngas heating value and cold gas efficiency. Analysis of the results shows the accuracy of the reduced order model to be similar to that of more detailed models that incorporate CFD. Next steps include the activation of pollutant chemistry and slag submodels, application of the reduced order model to other gasifier designs, parameter studies and uncertainty analysis of unknown and/or assumed physical and modeling parameters, and activation of dynamic simulation capability.

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.

Reduced-Order Modeling of Two-Sided Frictional Interfaces

2007 ◽  
Author(s):  
Jason D. Miller ◽  
D. Dane Quinn
Keyword(s):  
Modal Analysis ◽  
Computational Efficiency ◽  
Order Approximation ◽  
Reduced Order Modeling ◽  
Original Model ◽  
Reduced Order Model ◽  
Order Model ◽  
Reduced Order ◽  
Overall Response

We consider a model describing the behavior of a two-sided interface allowing for both elasticity and microslip of the joint. A reduced-order approximation of this system is developed based on a decomposition of the original model into an elastic chain and a dissipative component equivalent to a series-series Iwan chain. The Iwan chain is then solved using a quasi-static complementarity formulation while the order of the elastic chain is reduced using modal analysis. The computational efficiency of the resulting reduced-order model is significantly increased, while the overall response of the interface to realistic forcing conditions is maintained.

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.

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.

Reduced Order Modeling Based on Complex Nonlinear Modal Analysis and its Application to Bladed Disks With Shroud Contact

2013 ◽  
Author(s):  
Malte Krack ◽  
Lars Panning-von Scheidt ◽  
Jörg Wallaschek ◽  
Christian Siewert ◽  
Andreas Hartung
Keyword(s):  
Modal Analysis ◽  
Limit Cycle ◽  
Large Scale ◽  
Computational Effort ◽  
Reduced Order Modeling ◽  
Forced Response ◽  
Reduced Order Model ◽  
Order Model ◽  
Reduced Order ◽  
Nonlinear Modal Analysis

The design of bladed disks with contact interfaces typically requires analyses of the resonant forced response and flutter-induced limit cycle oscillations. The steady-state vibration behavior can efficiently be calculated using the Multi-Harmonic Balance method. The dimension of the arising algebraic systems of equations is essentially proportional to the number of harmonics and the number of degrees of freedom (DOFs) retained in the model. Extensive parametric studies necessary e.g. for robust design optimization are often not possible in practice due to the resulting computational effort. In this paper, a two-step nonlinear reduced order modeling approach is proposed. First, the autonomous nonlinear system is analyzed using a Complex Nonlinear Modal Analysis technique based on the work of Laxalde and Thouverez [1]. The methodology in [1] was refined by an exact condensation approach as well as analytical calculation of gradients in order to efficiently study localized nonlinearities in large-scale systems. Moreover, a continuation method was employed in order to predict nonlinear modal interactions. Modal properties such as eigenfrequency and modal damping are directly calculated with respect to the kinetic energy in the system. In a second step, a reduced order model is built based on the Single Nonlinear Resonant Mode theory. It is shown that linear damping and harmonic forcing can be superimposed. Moreover, similarity properties can be exploited to vary normal preload or gap values in contact interfaces. Thus, a large parameter space can be covered without the need for re-computation of nonlinear modal properties. The computational effort for evaluating the reduced order model is almost negligible since it contains a single DOF only, independent of the original system. The methodology is applied to both a simplified and a large-scale model of a bladed disk with shroud contact interfaces. In contrast to [1], the contact constraints account for variable normal load and lift-off in addition to dry friction. Forced response functions, backbone curves for varying normal preload and excitation level as well as flutter-induced limit cycle oscillations are analysed and compared to conventional methods. The limits of the proposed methodology are indicated and discussed.

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