A reduced order variational multiscale approach for turbulent flows

Stabile, Giovanni; Ballarin, Francesco; Zuccarino, Giacomo; Rozza, Gianluigi

doi:10.1007/s10444-019-09712-x

Cited by 79 publications

(60 citation statements)

References 52 publications

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“…Finally, we remark that alternative approaches to supremizer enrichment are available, see e.g. [80,34,81,79]. In particular, one that relies on the underlying P 1 /P 1 (Cut) FEM stabilization [80,81] seemed attractive in view of providing a less intrusive reduced order model, allowing to neglect the supremizer enrichment stage.…”

Section: Steady Stokes Problemmentioning

confidence: 99%

“…[80,34,81,79]. In particular, one that relies on the underlying P 1 /P 1 (Cut) FEM stabilization [80,81] seemed attractive in view of providing a less intrusive reduced order model, allowing to neglect the supremizer enrichment stage. However, numerical experiments carried out in the preparation of this work have shown a sensible deterioration of the accuracy when relying only on the high-fidelity stabilization without supremizer enrichment due to the weaker stabilization employed.…”

Section: Steady Stokes Problemmentioning

confidence: 99%

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Projection-based reduced order models for a cut finite element method in parametrized domains

Karatzas

Ballarin

Rozza

2020

Computers & Mathematics with Applications

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This work presents a reduced order modeling technique built on a high fidelity embedded mesh finite element method. Such methods, and in particular the CutFEM method, are attractive in the generation of projection-based reduced order models thanks to their capabilities to seamlessly handle large deformations of parametrized domains and in general to handle topological changes. The combination of embedded methods and reduced order models allows us to obtain fast evaluation of parametrized problems, avoiding remeshing as well as the reference domain formulation, often used in the reduced order modeling for boundary fitted finite element formulations. The resulting novel methodology is presented on linear elliptic and Stokes problems, together with several test cases to assess its capability. The role of a proper extension and transport of embedded solutions to a common background is analyzed in detail. Recent improvements go under the names of Ghost-Cell finite difference methods, Cut-Cell finite volume approach, Immersed Interface, Ghost Fluid, Volume Penalty methods, for which we refer to the review paper [1] and references within. In particular, for what concerns incompressible flows in arbitrary smooth domains, the Immersed Boundary Smooth Extension method has shown high-order convergence for the incompressible Navier-Stokes equations [4].More in detail, extended mesh finite element methods using cut elements are examined in [5,6] for stationary Stokes flow systems, as well as for Navier-Stokes. An analysis for high Reynolds numbers, independent of the local Reynolds, has been carried out in [7,8,9]. XFEM approaches in 2D and 3D Navier-Stokes are reported in [10]. Higher Reynolds number aerodynamics problems in unbounded domains, thin vortex ring and Lattice Green function Immersed Boundary methods are studied in [11,12]. Furthermore, embedded and immersed methods have been used in solving fluid structure

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Section: Steady Stokes Problemmentioning

confidence: 99%

Section: Steady Stokes Problemmentioning

confidence: 99%

Projection-based reduced order models for a cut finite element method in parametrized domains

Karatzas

Ballarin

Rozza

2020

Computers & Mathematics with Applications

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“…This is analogous to the closure problem encountered in large eddy simulation. Research has examined the construction of mixing length [28], Smagorinsky-type [27,29,30,31], and variational multiscale (VMS) closures [27,32,33,34] for POD-ROMs. The VMS approach is of particular relevance to this work.…”

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confidence: 99%

The Adjoint Petrov–Galerkin method for non-linear model reduction

Parish

Wentland

Duraisamy

2020

Computer Methods in Applied Mechanics and Engineering

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We formulate a new projection-based reduced-ordered modeling technique for non-linear dynamical systems. The proposed technique, which we refer to as the Adjoint Petrov-Galerkin (APG) method, is derived by decomposing the generalized coordinates of a dynamical system into a resolved coarse-scale set and an unresolved fine-scale set. A Markovian finite memory assumption within the Mori-Zwanzig formalism is then used to develop a reduced-order representation of the coarse-scales. This procedure leads to a closed reduced-order model that displays commonalities with the adjoint stabilization method used in finite elements. The formulation is shown to be equivalent to a Petrov-Galerkin method with a non-linear, time-varying test basis, thus sharing some similarities with the Least-Squares Petrov-Galerkin method. Theoretical analysis examining a priori error bounds and computational cost is presented. Numerical experiments on the compressible Navier-Stokes equations demonstrate that the proposed method can lead to improvements in numerical accuracy, robustness, and computational efficiency over the Galerkin method on problems of practical interest. Improvements in numerical accuracy and computational efficiency over the Least-Squares Petrov-Galerkin method are observed in most cases.(G ROM) has been used successfully in a variety of problems. When applied to general non-self-adjoint and non-linear problems, however, theoretical analysis and numerical experiments have shown that Galerkin ROM lacks a priori guarantees of stability, accuracy, and convergence [9]. This last issue is particularly challenging as it demonstrates that enriching a ROM basis does not necessarily improve the solution [10]. The development of stable and accurate reduced-order modeling techniques for complex non-linear systems is the motivation for the current work.A significant body of research aimed at producing accurate and stable ROMs for complex non-linear problems exists in the literature. These efforts include, but are not limited to, "energy-based" inner products [9,11], symmetry transformations [12], basis adaptation [13,14], L 1 -norm minimization [15], projection subspace rotations [16], and least-squares residual minimization approaches [17,18,19,20,21,22,23,24]. The Least-Squares Petrov-Galerkin (LSPG) [22] method comprises a particularly popular leastsquares residual minimization approach and has been proven to be an effective tool for non-linear model reduction. Defined at the fully-discrete level (i.e., after spatial and temporal discretization), LSPG relies on least-squares minimization of the FOM residual at each time-step. While the method lacks a priori stability guarantees for general non-linear systems, it has been shown to be effective for complex problems of interest [24,23,25]. Additionally, as it is formulated as a minimization problem, physical constraints such as conservation can be naturally incorporated into the ROM formulation [26]. At the fully-discrete level, LSPG is sensitive to both the time integration scheme as ...

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“…Further, a transport equation has been coupled with the incompressible Navier-Stokes equations and the corresponding modifications on the ROM have been presented. Besides, the good results obtained open the doors to the development of ROM for the t = 0.1s t = 0.5s t = 2.5s t = 5s resolution of turbulent flows by coupling the Navier-Stokes equations with the transport equations of a RANS or a LES turbulence model ( [34,39,43,41,27]).…”

Section: Conclusion and Future Developmentsmentioning

confidence: 91%

POD–Galerkin reduced order methods for combined Navier–Stokes transport equations based on a hybrid FV-FE solver

Busto

Stabile

Rozza

et al. 2020

Computers & Mathematics with Applications

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The purpose of this work is to introduce a novel POD-Galerkin strategy for the hybrid finite volume/finite element solver introduced in [9] and [11]. The interest is into the incompressible Navier-Stokes equations coupled with an additional transport equation. The full order model employed in this article makes use of staggered meshes. This feature will be conveyed to the reduced order model leading to the definition of reduced basis spaces in both meshes. The reduced order model presented herein accounts for velocity, pressure, and a transport-related variable. The pressure term at both the full order and the reduced order level is reconstructed making use of a projection method. More precisely, a Poisson equation for pressure is considered within the reduced order model. Results are verified against three-dimensional manufactured test cases. Moreover a modified version of the classical cavity test benchmark including the transport of a species is analysed.

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A reduced order variational multiscale approach for turbulent flows

Cited by 79 publications

References 52 publications

Projection-based reduced order models for a cut finite element method in parametrized domains

Projection-based reduced order models for a cut finite element method in parametrized domains

The Adjoint Petrov–Galerkin method for non-linear model reduction

POD–Galerkin reduced order methods for combined Navier–Stokes transport equations based on a hybrid FV-FE solver

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