Reduced‐order modeling based on POD of a parabolized Navier–Stokes equation model I: forward model

Du, Juan; Navon, I. M.; Steward, Jeffrey L.; Alekseev, A. K.; Luo, Zhendong

doi:10.1002/fld.2606

Cited by 29 publications

(17 citation statements)

References 27 publications

(30 reference statements)

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“…A unique aspect of this method is that the snapshot data is formed from the solution vectors containing angular coefficients. In addition these vectors are recorded through space, as opposed to time, and it is this approach that allows for time independent problems to be solved (which is similar to that of [23,24]). It has been shown that the angular discretised system of equations derived from the POD formulation can be constructed very efficiently.…”

Section: Resultsmentioning

confidence: 99%

“…(6). As no time dimension is available in which to take the snapshots, the set of snapshots are formed by recording these moments at different positions in space -in essence this work has interchanged the roles of space and time in standard POD, with angle and space, respectively (for similar techniques see [23,24]). The snapshots of angular moments are taken at each node defined by the full model's mesh.…”

Section: The Pod Angular Discretisationmentioning

confidence: 99%

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A POD reduced order model for resolving angular direction in neutron/photon transport problems

Buchan

Calloo

Goffin

et al. 2015

Journal of Computational Physics

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This article presents the first Reduced Order Model (ROM) that efficiently resolves the angular dimension of the time independent, mono-energetic Boltzmann Transport Equation (BTE). It is based on Proper Orthogonal Decomposition (POD) and uses the method of snapshots to form optimal basis functions for resolving the direction of particle travel in neutron/photon transport problems. A unique element of this work is that the snapshots are formed from the vector of angular coefficients relating to a high resolution expansion of the BTE's angular dimension. In addition, the individual snapshots are not recorded through time, as in standard POD, but instead they are recorded through space. In essence this work swaps the roles of the dimensions space and time in standard POD methods, with angle and space respectively. It is shown here how the POD model can be formed from the POD basis functions in a highly efficient manner. The model is then applied to two radiation problems; one involving the transport of radiation through a shield and the other through an infinite array of pins. Both problems are selected for their complex angular flux solutions in order to provide an appropriate demonstration of the model's capabilities. It is shown that the POD model can resolve these fluxes efficiently and accurately. In comparison to high resolution models this POD model can reduce the size of a problem by up to two orders of magnitude without compromising accuracy. Solving times are also reduced by similar factors.

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Section: Resultsmentioning

confidence: 99%

Section: The Pod Angular Discretisationmentioning

confidence: 99%

A POD reduced order model for resolving angular direction in neutron/photon transport problems

Buchan

Calloo

Goffin

et al. 2015

Journal of Computational Physics

Self Cite

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“…This method has been used to establish some POD-based reduced-order Galerkin, FE, and FD numerical models for PDEs (see [7,9,12,15,16,17,18]). Moreover, it has played an important role in the reduced-basis of numerical models for PDEs (see [10,19,21,24,25]).…”

Section: Introductionmentioning

confidence: 99%

A Pod-Based Reduced-Order Stabilized Crank–nicolson Mfe Formulation for the Non-Stationary Parabolized Navier–stokes Equations

Luo

2015

Mathematical Modelling and Analysis

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We firstly employ a proper orthogonal decomposition (POD) method, Crank-Nicolson (CN) technique, and two local Gaussian integrals to establish a PODbased reduced-order stabilized CN mixed finite element (SCNMFE) formulation with very few degrees of freedom for non-stationary parabolized Navier-Stokes equations. Then, the error estimates of the reduced-order SCNMFE solutions, which are acted as a suggestion for choosing number of POD basis and a criterion for updating POD basis, and the algorithm implementation for the POD-based reduced-order SCN-MFE formulation are provided, respectively. Finally, some numerical experiments are presented to illustrate that the numerical results are consistent with theoretical conclusions. Moreover, it is shown that the reduced-order SCNMFE formulation is feasible and efficient for finding numerical solutions of the non-stationary parabolized Navier-Stokes equations.Keywords: proper orthogonal decomposition method, reduced-order stabilized Crank-Nicolson mixed finite element formulation, non-stationary parabolized Navier-Stokes equations, error estimate.A POD-Based Reduced-Order SCNMFE Formulation 347 Problem I. Seek u = (u 1 , u 2 ) T and p satisfying for T ∞ > 0,where Ω ⊂ R 2 is a bounded and connected domain, ∂ t = ∂/∂t represents the first order partial derivative with respect to time t, ∂ 2 yy = ∂ 2 /∂y 2 the second order partial derivative with respect to y, u = (u 1 , u 2 ) T the velocity vector, p the pressure, T ∞ the total time, ν = (RePr ) −1 , Re the Reynolds number, Pr the Prandtl number, and ϕ(x, y, t) and u 0 (x, y) are given vector functions. As a matter of convenience and without loss of generality, we might assume that ϕ(x, y, t) = 0 in the following theoretical analysis.A lot of numerical examples have shown that, if the fluid mainstream direction does not appear on a wide range of separation zone, the numerical results for the simplified parabolized Navier-Stokes equations are very close to those for the full Navier-Stokes equations. Especially, for fluid flow with high Reynolds number, the numerical viscosity for the full Navier-Stokes equations tends to hide some of the real physical viscosity. In other word, for fluid flow problems with high Reynolds number, the numerical solutions obtained from the parabolized Navier-Stokes equations are closer to real physical solutions than those obtained from the full Navier-Stokes equations (see [2]). Although, it seems, in principle, unreasonable that the fluid flow of the computational domain appearing separation zone is described by the simplified parabolized Navier-Stokes equations, a lot of computational examples show that, for high Reynolds number fluid flows, which feature a local small separation zone along the main stream direction (for example, the front or backward facing step flow, the separation bubble flow, the compression corner flow, the air intake channel flow field), the numerical solutions obtained by the simplified parabolized Navier-Stokes equations are very close to those obtained from the full Nav...

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“…Finally, it is also important to note that this approach has similarities with that of [27], which solved the two dimensional time independent parabolized Navier-Stokes equation. In this previous work snapshots were constructed by solving the problem using one of the spatial dimensions as though it were time.…”

Section: Introductionmentioning

confidence: 99%

“…It has been applied to the shallow water equations [24], the Euler equations [25], the full Navier Stokes equations [26] and the various reduced versions of it, e.g. the parabolized Navier Stokes [27,28].…”

Section: Introductionmentioning

confidence: 99%

A POD reduced‐order model for eigenvalue problems with application to reactor physics

Buchan

Pain

Fang

et al. 2013

Numerical Meth Engineering

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SUMMARYA reduced order model (ROM) based on proper orthogonal decomposition (POD) has been presented and applied to solving eigenvalue problems. The model is constructed via the method of snapshots which is based upon the singular value decomposition of a matrix containing the characteristics of a solution as it evolves through time. Part of the novelty of this work is in how this snapshot data is generated, and this is through the recasting of eigenvalue problem, which is time-independent, into a time dependent form.Instances of time dependent eigenfunction solutions are therefore used to construct the snapshot matrix.The reduced order model's capabilities in efficiently resolving eigenvalue problems that typically become computationally expensive (using standard full model discretisations) has been demonstrated. Whilst the approach can be adapted to most general eigenvalue problems, the examples presented here are based on calculating dominant eigenvalues in reactor physics applications. The approach is shown to reconstruct both the eigenvalues and eigenfunctions accurately using a significantly reduced number of unknowns in comparison to 'full' models based on finite element discretisations. The novelty of this paper therefore includes a new approach to generating snapshots, POD's application to large scale eigenvalue calculations, and ROM's application in reactor physics.

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Reduced‐order modeling based on POD of a parabolized Navier–Stokes equation model I: forward model

Cited by 29 publications

References 27 publications

A POD reduced order model for resolving angular direction in neutron/photon transport problems

A POD reduced order model for resolving angular direction in neutron/photon transport problems

A Pod-Based Reduced-Order Stabilized Crank–nicolson Mfe Formulation for the Non-Stationary Parabolized Navier–stokes Equations

A POD reduced‐order model for eigenvalue problems with application to reactor physics

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