An Evolve-Filter-Relax Stabilized Reduced Order Stochastic Collocation Method for the Time-Dependent Navier--Stokes Equations

We present a stabilized POD–Galerkin reduced order method (ROM) for a Leray model. For the implementation of the model, we combine a two-step algorithm called Evolve-Filter (EF) with a computationally efficient finite volume method. In both steps of the EF algorithm, velocity and pressure fields are approximated using different POD basis and coefficients. To achieve pressure stabilization, we consider and compare two strategies: the pressure Poisson equation and the supremizer enrichment of the velocity space. We show that the evolve and filtered velocity spaces have to be enriched with the supremizer solutions related to both evolve and filter pressure fields in order to obtain stable and accurate solutions with the supremizer enrichment method. We test our ROM approach on a 2D unsteady flow past a cylinder at Reynolds number 0≤Re≤100. We find that both stabilization strategies produce comparable errors in the reconstruction of the lift and drag coefficients, with the pressure Poisson equation method being more computationally efficient.

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“…Note that Φ could be either a scalar or a vector field. Using (24) to approximate v n+1 and q n+1 in ( 9) and ( 10), we obtain…”

Section: A Pod-galerkin Projection Methodsmentioning

confidence: 99%

“…In the context of regularized ROMs, the Leray model and the EF algorithm have been thoroughly investigated. See, e.g., [21][22][23][24][25]. In all of these works, the filtering approach is only employed at reduced order level.…”

Section: Introductionmentioning

confidence: 99%

Pressure Stabilization Strategies for a LES Filtering Reduced Order Model

Girfoglio

Quaini

Rozza

2021

Fluids

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“…However, this truncation ignores the dyadic interactions between the first R modes and the remaining ones. As a result, an erroneous behavior might arise in the ROM solution [99][100][101][102], and closure and stabilization techniques have been introduced to improve ROM accuracy [71,84,[103][104][105]. As highlighted earlier in Sec.…”

Section: D Kraichnan Turbulencementioning

confidence: 99%

Forward sensitivity approach for estimating eddy viscosity closures in nonlinear model reduction

et al. 2020

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In this paper, we propose a variational approach to estimate eddy viscosity using forward sensitivity method (FSM) for closure modeling in nonlinear reduced order models. FSM is a data assimilation technique that blends model's predictions with noisy observations to correct initial state and/or model parameters. We apply this approach on a projection based reduced order model (ROM) of the one-dimensional viscous Burgers equation with a square wave defining a moving shock, and the two-dimensional vorticity transport equation formulating a decay of Kraichnan turbulence. We investigate the capability of the approach to approximate an optimal value for eddy viscosity with different measurement configurations. Specifically, we show that our approach can sufficiently assimilate information either through full-field or sparse noisy measurements to estimate eddy viscosity closure to cure standard Galerkin reduced order model (GROM) predictions. Therefore, our approach provides a modular framework to correct forecasting error from a sparse observational network on a latent space. We highlight that the proposed GROM-FSM framework is promising for emerging digital twin applications, where real-time sensor measurements can be used to update and optimize surrogate model's parameters.

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“…Thus, such closure terms usually involve empirical choices of the structure and parameters that prevent the generalization. Filter‐based stabilization of ROMs has also been pursued in several works, 60‐62 where different types of filters are developed to alleviate numerical oscillations and stabilize Galerkin ROMs in convection‐dominated flows.…”

Section: Introductionmentioning

confidence: 99%

A global eigenvalue reassignment method for the stabilization of nonlinear reduced‐order models

Rezaian

Wei

2021

Numerical Meth Engineering

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The systematic and physics‐infused construction of a projection‐based reduced‐order model (ROM) shows the capability to replicate the original high‐dimensional system's dynamical evolution but with a fractional computational cost. However, certain nonlinear features and high‐frequency contributions may be lost throughout the aggressive order reduction. Thus, ROMs in a broad category of fluid dynamics applications require stabilization and closure methods to compensate for the key contributions missed through the model order reduction. A new stabilization method for nonlinear ROMs is proposed in this study that learns a linear control law to drive the nonlinear ROM toward maximum agreement with the full‐order model, where a total power constraint guarantees the stability of the nonlinear ROM. This new method achieves both stability and accuracy for nonlinear proper orthogonal decomposition‐Galerkin ROMs in two chosen applications with strong shock‐wake interactions and unsteady oscillations, which trigger strong instabilities in the original ROMs before stabilization. A multistage layout is designed to further enhance the proposed stabilization method for more efficient and robust stabilization of nonlinear ROMs with a large number of unstable modes.

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An Evolve-Filter-Relax Stabilized Reduced Order Stochastic Collocation Method for the Time-Dependent Navier--Stokes Equations

Cited by 25 publications

References 42 publications

Pressure Stabilization Strategies for a LES Filtering Reduced Order Model

Pressure Stabilization Strategies for a LES Filtering Reduced Order Model

Forward sensitivity approach for estimating eddy viscosity closures in nonlinear model reduction

A global eigenvalue reassignment method for the stabilization of nonlinear reduced‐order models

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