On the stability and extension of reduced-order Galerkin models in incompressible flows

Akhtar, Imran; Nayfeh, Ali H.; Ribbens, Calvin J.

doi:10.1007/s00162-009-0112-y

Cited by 183 publications

(155 citation statements)

References 41 publications

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“…Respect to what presented in [12] here also the gradient of pressure is considered inside the momentum equation. This term is considered also in [13,14] but, there, it is assumed that velocity and pressure modes share the same temporal coefficients.…”

Section: Rom For Velocity -Momentum Equationmentioning

confidence: 99%

“…Such approach, that is proposed in literature by several authors [12,24] for FEM approximations, is here adapted to a FVM framework.…”

Section: The Reduced Order Modelmentioning

confidence: 99%

“…The reduced order model of the momentum equation is obtained performing an L 2 orthogonal projection onto the reduced bases space V P OD spanned by the POD velocity modes with a procedure similar to what presented in [12].…”

Section: Rom For Velocity -Momentum Equationmentioning

confidence: 99%

“…In this method the governing equations, describing the phenomenon, are projected onto a low dimensional space called the reduced basis space [9,10] that is optimally constructed starting from high fidelity simulations. In particular in this work the ROM is constructed using a POD-Galerkin approach [11][12][13][14][15][16][17]. As previously mentioned, since the main purpose of this paper is the correct modelling of the vortex shedding phenomenon, particular attention is paid to the reconstruction in the reduced order model of the pressure field.…”

Section: Introductionmentioning

confidence: 99%

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

Stabile

Hijazi

Mola

et al. 2017

Communications in Applied and Industrial Mathematics

142

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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 flow 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 pressure 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.

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Section: Rom For Velocity -Momentum Equationmentioning

confidence: 99%

“…Such approach, that is proposed in literature by several authors [12,24] for FEM approximations, is here adapted to a FVM framework.…”

Section: The Reduced Order Modelmentioning

confidence: 99%

Section: Rom For Velocity -Momentum Equationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

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

Stabile

Hijazi

Mola

et al. 2017

Communications in Applied and Industrial Mathematics

142

View full text Add to dashboard Cite

show abstract

“…The reduced-order model equations are completed with a model for the pressure, which is derived using a discrete pressure Poisson equation [3] and the pressure reduced basis vectors [4]. The projected equations are given as…”

Section: Reduced-order Modelingmentioning

confidence: 99%

POD and CVT Galerkin reduced‐order modeling of the flow around a cylinder

Ullmann

Lang

2012

Proc Appl Math and Mech

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Reduced-order models are applied to the laminar vortex-shedding flow around a circular cylinder. The models rely on approximations of the solutions in the space spanned by a set of POD or CVT reduced basis vectors, which are computed from snapshots of a numerical solution. To enable the computation of drag and lift forces, the modeling of the velocity and pressure is realized via a one-way coupling. A comparison of the results of the CVT and POD reduced-order models is presented. Reduced-order modelingA spatially semi-discretized incompressible Navier-Stokes problem can be obtained by inserting functions of a finite element space in the weak form of the governing equations. The resulting finite element model is given bywhere the vectors U (t) and P (t) contain the values of the velocity and pressure at the mesh nodes, and U (0) = U 0 . Reducedorder models can be derived in an analogical way by inserting functions of a reduced-order subspace in the weak form or, alternatively, by projecting the finite element model on the space spanned by a set of reduced basis vectors. Velocity reduced basis vectors Φ 1 , . . . , Φ R are computed from a set of snapshots U 1 , . . . , U N , where typically R ≪ N . The proper orthogonal decomposition (POD) [1] generates basis vectors by solving the minimization problemwhere M indicates norms and inner products induced by the finite element mass matrix. The solution of the minimization problem can be found by singular value decomposition. The centroidal Voronoi tessellation (CVT) assigns each snapshot to one of the clusters V 1 , . . . , V R and defines Φ 1 , . . . , Φ R as the cluster centers which solveSolutions of this minimization problem can be approximated in an iterative way using Lloyd's method [2]. Reduced basis vectors Φ 1 , . . . , Φ R are now computed usingwhere U ref is a reference velocity. Additionally, reduced basis vectors Ψ 1 , . . . , Ψ R are computed from a set of pressure snapshots P 1 , . . . , P N . The discrete velocity and pressure fields can then be approximated usingFor the velocity reduced-order model, a Galerkin projection of (1) on the space spanned by the velocity reduced basis vectors is performed, which gives Φ T r M˙ U R = Φ T r N ( U R ) U R + Φ T r K U R + Φ T r CP, r = 1, . . . , R.

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An evolve‐then‐filter regularized reduced order model for convection‐dominated flows

Wells

Wang

Xie

et al. 2017

Numerical Methods in Fluids

118

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Summary In this paper, we propose a new evolve‐then‐filter reduced order model (EF‐ROM). This is a regularized ROM (Reg‐ROM), which aims to add numerical stabilization to proper orthogonal decomposition (POD) ROMs for convection‐dominated flows. We also consider the Leray ROM (L‐ROM). These two Reg‐ROMs use explicit ROM spatial filtering to smooth (regularize) various terms in the ROMs. Two spatial filters are used: a POD projection onto a POD subspace (Proj) and a POD differential filter (DF). The four Reg‐ROM/filter combinations are tested in the numerical simulation of the three‐dimensional flow past a circular cylinder at a Reynolds number Re=1000. Overall, the most accurate Reg‐ROM/filter combination is EF‐ROM‐DF. Furthermore, the spatial filter has a higher impact on the Reg‐ROM than the regularization used. Indeed, the DF generally yields better results than Proj for both the EF‐ROM and L‐ROM. Finally, the CPU times of the four Reg‐ROM/filter combinations are orders of magnitude lower than the CPU time of the DNS. Copyright © 2017 John Wiley & Sons, Ltd.

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On the stability and extension of reduced-order Galerkin models in incompressible flows

Cited by 183 publications

References 41 publications

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

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

POD and CVT Galerkin reduced‐order modeling of the flow around a cylinder

An evolve‐then‐filter regularized reduced order model for convection‐dominated flows

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