Finite volume POD-Galerkin stabilised reduced order methods for the parametrised incompressible Navier–Stokes equations

Stabile, Giovanni; Rozza, Gianluigi

doi:10.1016/j.compfluid.2018.01.035

Cited by 179 publications

(239 citation statements)

References 59 publications

(107 reference statements)

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“…Then, more modes can be used to identify the reduced matrix with the least‐squares technique as, in order to keep the system overdetermined, the size of the reduced matrix is limited by the square root of the number of snapshots. Even more snapshots would be required in the case of nonlinear systems because then at least as many reduced matrices are to be identified as there are modes to be stored in the offline phase . For example, the nonlinear convective term of the Navier‐Stokes equations can be approximated by

{bold-italica}^{T} {bold-italicC}_{r} bold-italica

, where C r is a third‐order tensor.…”

Section: Discussionmentioning

confidence: 99%

“…Even more snapshots would be required in the case of nonlinear systems because then at least as many reduced matrices are to be identified as there are modes to be stored in the offline phase. [24][25][26] For example, the nonlinear convective term of the Navier-Stokes equations can be approximated by a T C r a, where C r is a third-order tensor. Then, the reduced problem grows with the cube of the number of modes in order to maintain an offline-online decomposition.…”

Section: Discussionmentioning

confidence: 99%

“…As the POD modes are orthogonal to each other,

{⟨ {bold-italicϕ}_{i}, {bold-italicϕ}_{j} ⟩}_{L_{2} false(normalΩ false)} = δ_{i j}

, the POD basis, E POD , is optimal when the difference between the snapshots and the projection of the snapshots on the basis functions is minimal for a certain norm. The L 2 ‐norm is preferred for discrete numerical schemes with

{⟨ \cdot, \cdot ⟩}_{L_{2} false(normalΩ false)}

the L 2 inner product of the functions over the domain Ω. The minimization problem is given by

E_{N_{r}}^{P O D} = arg min \frac{1}{N_{t} + 1} true \sum_{n = 0}^{N_{t}} {(‖‖, T false(bold-italicx, t^{n} false) - true \sum_{i = 1}^{N_{r}} {(⟨⟩, T false(bold-italicx, t^{n} false), ϕ_{i} false(bold-italicx false))}_{L_{2}} ϕ_{i} false(bold-italicx false))}_{L_{2}}^{2} .

…”

Section: The Rommentioning

confidence: 99%

“…As the POD modes are orthogonal to each other, ⟨ i , ⟩ L 2 (Ω) = i , the POD basis, E POD , is optimal when the difference between the snapshots and the projection of the snapshots on the basis functions is minimal for a certain norm. The L 2 -norm is preferred for discrete numerical schemes 24 with ⟨·, ·⟩ L 2 (Ω) the L 2 inner product of the functions over the domain Ω. The minimization problem is given by…”

Section: Generation Of the Pod Basismentioning

confidence: 99%

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POD‐identification reduced order model of linear transport equations for control purposes

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Belloni

Eynde

et al. 2019

Numerical Methods in Fluids

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Summary Intrusive reduced order modeling techniques require access to the solver's discretization and solution algorithm, which are not available for most computational fluid dynamics codes. Therefore, a nonintrusive reduction method that identifies the system matrix of linear fluid dynamical problems with a least‐squares technique is presented. The methodology is applied to the linear scalar transport convection‐diffusion equation for a 2D square cavity problem with a heated lid. The (time‐dependent) boundary conditions are enforced in the obtained reduced order model (ROM) with a penalty method. The results are compared and the accuracy of the ROMs is assessed against the full order solutions and it is shown that the ROM can be used for sensitivity analysis by controlling the nonhomogeneous Dirichlet boundary conditions.

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{bold-italica}^{T} {bold-italicC}_{r} bold-italica

, where C r is a third‐order tensor.…”

Section: Discussionmentioning

confidence: 99%

Section: Discussionmentioning

confidence: 99%

“…As the POD modes are orthogonal to each other,

{⟨ {bold-italicϕ}_{i}, {bold-italicϕ}_{j} ⟩}_{L_{2} false(normalΩ false)} = δ_{i j}

{⟨ \cdot, \cdot ⟩}_{L_{2} false(normalΩ false)}

the L 2 inner product of the functions over the domain Ω. The minimization problem is given by

E_{N_{r}}^{P O D} = arg min \frac{1}{N_{t} + 1} true \sum_{n = 0}^{N_{t}} {(‖‖, T false(bold-italicx, t^{n} false) - true \sum_{i = 1}^{N_{r}} {(⟨⟩, T false(bold-italicx, t^{n} false), ϕ_{i} false(bold-italicx false))}_{L_{2}} ϕ_{i} false(bold-italicx false))}_{L_{2}}^{2} .

…”

Section: The Rommentioning

confidence: 99%

Section: Generation Of the Pod Basismentioning

confidence: 99%

See 2 more Smart Citations

POD‐identification reduced order model of linear transport equations for control purposes

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Belloni

Eynde

et al. 2019

Numerical Methods in Fluids

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“…The first type of instability, that is associated with the fact of considering both velocity and pressure terms at the reduced order level, can be resolved using a variety of different numerical techniques. The supremizer stabilisation method [28,3,33], a stabilisation based on the use of a Poisson equation for pressure [32,2] or Petrov-Galerkin projection strategies [8,9] stand out among others. Along these lines, it is a common practice to neglect the pressure contribution and to generate a reduced order model that accounts for the velocity field only.…”

Section: Introductionmentioning

confidence: 99%

Bayesian identification of a projection-based reduced order model for computational fluid dynamics

Stabile

Rosić

2020

Computers & Fluids

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In this paper we propose a Bayesian method as a numerical way to correct and stabilise projection-based reduced order models (ROM) in computational fluid dynamics problems. The approach is of hybrid type, and consists of the classical proper orthogonal decomposition driven Galerkin projection of the laminar part of the governing equations, and Bayesian identification of the correction term mimicking both the turbulence model and possible ROM-related instabilities given the full order data. In this manner the classical ROM approach is translated to the parameter identification problem on a set of nonlinear ordinary differential equations. Computationally the inverse problem is solved with the help of the Gauss-Markov-Kalman smoother in both ensemble and square-root polynomial chaos expansion forms. To reduce the dimension of the posterior space, a novel global variance based sensitivity analysis is proposed.

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Reduced order models for the incompressible Navier‐Stokes equations on collocated grids using a ‘discretize‐then‐project’ approach

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Sanderse

Stabile

et al. 2021

Numerical Methods in Fluids

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A novel reduced order model (ROM) for incompressible flows is developed by performing a Galerkin projection based on a fully (space and time) discrete full order model (FOM) formulation. This 'discretize-then-project' approach requires no pressure stabilization technique (even though the pressure term is present in the ROM) nor a boundary control technique (to impose the boundary conditions at the ROM level). These are two main advantages compared to existing approaches. The fully discrete FOM is obtained by a finite volume discretization of the incompressible Navier-Stokes equations on a collocated grid, with a forward Euler time discretization. Two variants of the time discretization method, the inconsistent and consistent flux method, have been investigated. The latter leads to divergence-free velocity fields, also on the ROM level, whereas the velocity fields are only approximately divergencefree in the former method. For both methods, stable and accurate results have been obtained for test cases with different types of boundary conditions: a lid-driven cavity and an open-cavity (with an inlet and outlet). The ROM obtained with the consistent flux method, having divergence-free velocity fields, is slightly more accurate but also slightly more expensive to solve compared to the inconsistent flux method. The speedup ratio of the ROM and FOM computation times is the highest for the open cavity test case with the inconsistent flux method.

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Finite volume POD-Galerkin stabilised reduced order methods for the parametrised incompressible Navier–Stokes equations

Cited by 179 publications

References 59 publications

POD‐identification reduced order model of linear transport equations for control purposes

POD‐identification reduced order model of linear transport equations for control purposes

Bayesian identification of a projection-based reduced order model for computational fluid dynamics

Reduced order models for the incompressible Navier‐Stokes equations on collocated grids using a ‘discretize‐then‐project’ approach

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