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Reduced Order Models for Computational Fluid Dynamics

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A special issue of Fluids (ISSN 2311-5521). This special issue belongs to the section "Mathematical and Computational Fluid Mechanics".

Deadline for manuscript submissions: closed (30 June 2021) | Viewed by 23387

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Special Issue Editor

Dr. Giovanni Stabile

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Guest Editor

SISSA, International School for Advanced Studies, Mathematics Area, MathLab, Trieste, Italy
Interests: reduced-order modeling for fluids; machine learning methods for fluid simulations; fluid-structure interaction; uncertainty quantification

Special Issue Information

Dear Colleagues,

Despite the recent increase in the available computation power, there are still several cases in computational fluid dynamics where standard discretization techniques (finite elements, finite volumes, finite differences, spectral elements, etc.) become unaffordable. Such situations occur when a large number of system configurations are in need of being tested, or limited computational time is required. Typical examples of this type are shape optimization, uncertainty quantification, and real-time control. A viable approach to reduce the computational burden is given by reduced-order models. Many different types of reduced order have been developed over the years, and a possible distinction is between those that are intrusive and require the knowledge of the underlying full order model and those that are merely data-driven and therefore non-intrusive. In the first category fall the reduced-basis method, the POD-Galerkin approach, and the proper generalized decomposition. In the second category, one can find truncation-based methods, dynamic mode decomposition, neural networks, and in general all the models based on just input–output data. Possible applications include but are not limited to uncertainty quantification, inverse problems, real-time control, shape optimization, etc.

This Special Issue will publish original research, overviews, and applications on reduced-order models for computational fluid dynamics of both the intrusive and non-intrusive type.

Dr. Giovanni Stabile
Guest Editor

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Keywords

Certified reduced basis method
Proper orthogonal decomposition
Dynamic mode decomposition
Data-driven modeling
Hyper-reduction techniques
Uncertainty quantification
Closure modeling
Machine learning for fluids

Published Papers (10 papers)

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Research

30 pages, 34456 KiB

Open AccessArticle

Adaptive Data-Driven Model Order Reduction for Unsteady Aerodynamics

by Peter Nagy and Marco Fossati

Fluids 2022, 7(4), 130; https://0-doi-org.brum.beds.ac.uk/10.3390/fluids7040130 - 06 Apr 2022

Cited by 3 | Viewed by 2017

Abstract

A data-driven adaptive reduced order modelling approach is presented for the reconstruction of impulsively started and vortex-dominated flows. A residual-based error metric is presented for the first time in the framework of the adaptive approach. The residual-based adaptive Reduced Order Modelling selects locally in time the most accurate reduced model approach on the basis of the lowest residual produced by substituting the reconstructed flow field into a finite volume discretisation of the Navier–Stokes equations. A study of such an error metric was performed to assess the performance of the resulting residual-based adaptive framework with respect to a single-ROM approach based on the classical proper orthogonal decomposition, as the number of modes is varied. Two- and three-dimensional unsteady flows were considered to demonstrate the key features of the method and its performance. Full article

(This article belongs to the Special Issue Reduced Order Models for Computational Fluid Dynamics)

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13 pages, 9356 KiB

Open AccessArticle

Efficient Reduced Order Modeling of Large Data Sets Obtained from CFD Simulations

by Thomas Holemans, Zhu Yang and Maarten Vanierschot

Fluids 2022, 7(3), 110; https://0-doi-org.brum.beds.ac.uk/10.3390/fluids7030110 - 17 Mar 2022

Cited by 3 | Viewed by 2328

Abstract

The ever-increasing computational power has shifted direct numerical simulations towards higher Reynolds numbers and large eddy simulations towards industrially-relevant flow scales. However, this increase in both temporal and spatial resolution has severely increased the computational cost of model order reduction techniques. Reducing the full data set to a smaller subset in order to perform reduced-order modeling (ROM) may be an interesting method to keep the computational effort reasonable. Moreover, non-tomographic particle image velocimetry measurements obtain a 2D data set of a 3D flow field and an interesting research question would be to quantify the difference between this 2D ROM compared to the 3D ROM of the full flow field. To provide an answer to both issues, the aim of this study was to test a new method for obtaining POD basis functions from a small subset of data initially and using them afterwards in the ROM of either the complete data set or the reduced data set. Hence, no new method of ROM is presented, but we demonstrate a procedure to significantly reduce the computational effort required for the ROM of very large data sets and a quantification of the error introduced by reducing the size of those data sets. The method applies eigenvalue decomposition on a small subset of data extracted from a full 3D simulation and the obtained temporal coefficients are projected back on the 3D velocity fields to obtain the 3D spatial modes. To test the method, an annular jet was chosen as a flow topology due to its simple geometry and the rich dynamical content of its flow field. First, a smaller data set is extracted from the 2D cross-sectional planes and ROM is performed on that data set. Secondly, the full 3D spatial structures are reconstructed by projecting the temporal coefficients back on the 3D velocity fields and the 2D spatial structures by projecting the temporal coefficients back on the 2D velocity fields. It is shown that two perpendicular lateral planes are sufficient to capture the relevant large-scale structures. As such, the total processing time can be reduced by a factor of 136 and up to 22 times less RAM is needed to complete the ROM processing. Full article

(This article belongs to the Special Issue Reduced Order Models for Computational Fluid Dynamics)

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21 pages, 8024 KiB

Open AccessArticle

Data-Targeted Prior Distribution for Variational AutoEncoder

by Nissrine Akkari, Fabien Casenave, Thomas Daniel and David Ryckelynck

Fluids 2021, 6(10), 343; https://0-doi-org.brum.beds.ac.uk/10.3390/fluids6100343 - 29 Sep 2021

Cited by 3 | Viewed by 2062

Abstract

Bayesian methods were studied in this paper using deep neural networks. We are interested in variational autoencoders, where an encoder approaches the true posterior and the decoder approaches the direct probability. Specifically, we applied these autoencoders for unsteady and compressible fluid flows in aircraft engines. We used inferential methods to compute a sharp approximation of the posterior probability of these parameters with the transient dynamics of the training velocity fields and to generate plausible velocity fields. An important application is the initialization of transient numerical simulations of unsteady fluid flows and large eddy simulations in fluid dynamics. It is known by the Bayes theorem that the choice of the prior distribution is very important for the computation of the posterior probability, proportional to the product of likelihood with the prior probability. Hence, we propose a new inference model based on a new prior defined by the density estimate with the realizations of the kernel proper orthogonal decomposition coefficients of the available training data. We numerically show that this inference model improves the results obtained with the usual standard normal prior distribution. This inference model was constructed using a new algorithm improving the convergence of the parametric optimization of the encoder probability distribution that approaches the posterior. This latter probability distribution is data-targeted, similarly to the prior distribution. This new generative approach can also be seen as an improvement of the kernel proper orthogonal decomposition method, for which we do not usually have a robust technique for expressing the pre-image in the input physical space of the stochastic reduced field in the feature high-dimensional space with a kernel inner product. Full article

(This article belongs to the Special Issue Reduced Order Models for Computational Fluid Dynamics)

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22 pages, 3771 KiB

Open AccessArticle

Neural Network-Based Model Reduction of Hydrodynamics Forces on an Airfoil

by Hamayun Farooq, Ahmad Saeed, Imran Akhtar and Zafar Bangash

Fluids 2021, 6(9), 332; https://0-doi-org.brum.beds.ac.uk/10.3390/fluids6090332 - 15 Sep 2021

Cited by 8 | Viewed by 2293

Abstract

In this paper, an artificial neural network (ANN)-based reduced order model (ROM) is developed for the hydrodynamics forces on an airfoil immersed in the flow field at different angles of attack. The proper orthogonal decomposition (POD) of the flow field data is employed to obtain pressure modes and the temporal coefficients. These temporal pressure coefficients are used to train the ANN using data from three different angles of attack. The trained network then takes the value of angle of attack (AOA) and past POD coefficients as an input and predicts the future temporal coefficients. We also decompose the surface pressure modes into lift and drag components. These surface pressure modes are then employed to calculate the pressure component of lift

C_{L}^{p}

and drag

C_{D}^{p}

coefficients. The train model is then tested on the in-sample data and out-of-sample data. The results show good agreement with the true numerical data, thus validating the neural network based model. Full article

(This article belongs to the Special Issue Reduced Order Models for Computational Fluid Dynamics)

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16 pages, 1926 KiB

Open AccessArticle

Pressure Stabilization Strategies for a LES Filtering Reduced Order Model

by Michele Girfoglio, Annalisa Quaini and Gianluigi Rozza

Fluids 2021, 6(9), 302; https://0-doi-org.brum.beds.ac.uk/10.3390/fluids6090302 - 25 Aug 2021

Cited by 10 | Viewed by 1860

Abstract

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 \leq R e \leq 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. Full article

(This article belongs to the Special Issue Reduced Order Models for Computational Fluid Dynamics)

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21 pages, 3210 KiB

Open AccessArticle

Online Coupled Generalized Multiscale Finite Element Method for the Poroelasticity Problem in Fractured and Heterogeneous Media

by Aleksei Tyrylgin, Maria Vasilyeva, Dmitry Ammosov, Eric T. Chung and Yalchin Efendiev

Fluids 2021, 6(8), 298; https://0-doi-org.brum.beds.ac.uk/10.3390/fluids6080298 - 23 Aug 2021

Cited by 4 | Viewed by 1702

Abstract

In this paper, we consider the poroelasticity problem in fractured and heterogeneous media. The mathematical model contains a coupled system of equations for fluid pressures and displacements in heterogeneous media. Due to scale disparity, many approaches have been developed for solving detailed fine-grid problems on a coarse grid. However, some approaches can lack good accuracy on a coarse grid and some corrections for coarse-grid solutions are needed. In this paper, we present a coarse-grid approximation based on the generalized multiscale finite element method (GMsFEM). We present the construction of the offline and online multiscale basis functions. The offline multiscale basis functions are precomputed for the given heterogeneity and fracture network geometry, where for the construction, we solve a local spectral problem and use the dominant eigenvectors (appropriately defined) to construct multiscale basis functions. To construct the online basis functions, we use current information about the local residual and solve coupled poroelasticity problems in local domains. The online basis functions are used to enrich the offline multiscale space and rapidly reduce the error using residual information. Only with appropriate offline coarse-grid spaces can one guarantee a fast convergence of online methods. We present numerical results for poroelasticity problems in fractured and heterogeneous media. We investigate the influence of the number of offline and online basis functions on the relative errors between the multiscale solution and the reference (fine-scale) solution. Full article

(This article belongs to the Special Issue Reduced Order Models for Computational Fluid Dynamics)

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24 pages, 3072 KiB

Open AccessArticle

Hybrid Neural Network Reduced Order Modelling for Turbulent Flows with Geometric Parameters

by Matteo Zancanaro, Markus Mrosek, Giovanni Stabile, Carsten Othmer and Gianluigi Rozza

Fluids 2021, 6(8), 296; https://0-doi-org.brum.beds.ac.uk/10.3390/fluids6080296 - 22 Aug 2021

Cited by 11 | Viewed by 2300

Abstract

Geometrically parametrized partial differential equations are currently widely used in many different fields, such as shape optimization processes or patient-specific surgery studies. The focus of this work is some advances on this topic, capable of increasing the accuracy with respect to previous approaches while relying on a high cost–benefit ratio performance. The main scope of this paper is the introduction of a new technique combining a classical Galerkin-projection approach together with a data-driven method to obtain a versatile and accurate algorithm for the resolution of geometrically parametrized incompressible turbulent Navier–Stokes problems. The effectiveness of this procedure is demonstrated on two different test cases: a classical academic back step problem and a shape deformation Ahmed body application. The results provide insight into details about the properties of the architecture we developed while exposing possible future perspectives for this work. Full article

(This article belongs to the Special Issue Reduced Order Models for Computational Fluid Dynamics)

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27 pages, 969 KiB

Open AccessFeature PaperArticle

Efficient Wildland Fire Simulation via Nonlinear Model Order Reduction

by Felix Black, Philipp Schulze and Benjamin Unger

Fluids 2021, 6(8), 280; https://0-doi-org.brum.beds.ac.uk/10.3390/fluids6080280 - 11 Aug 2021

Cited by 4 | Viewed by 1841

Abstract

We propose a new hyper-reduction method for a recently introduced nonlinear model reduction framework based on dynamically transformed basis functions and especially well-suited for transport-dominated systems. Furthermore, we discuss applying this new method to a wildland fire model whose dynamics feature traveling combustion waves and local ignition and is thus challenging for classical model reduction schemes based on linear subspaces. The new hyper-reduction framework allows us to construct parameter-dependent reduced-order models (ROMs) with efficient offline/online decomposition. The numerical experiments demonstrate that the ROMs obtained by the novel method outperform those obtained by a classical approach using the proper orthogonal decomposition and the discrete empirical interpolation method in terms of run time and accuracy. Full article

(This article belongs to the Special Issue Reduced Order Models for Computational Fluid Dynamics)

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22 pages, 4572 KiB

Open AccessArticle

Data-Driven Reduced-Order Modeling of Convective Heat Transfer in Porous Media

by Péter German, Mauricio E. Tano, Carlo Fiorina and Jean C. Ragusa

Fluids 2021, 6(8), 266; https://0-doi-org.brum.beds.ac.uk/10.3390/fluids6080266 - 28 Jul 2021

Cited by 2 | Viewed by 2945

Abstract

This work presents a data-driven Reduced-Order Model (ROM) for parametric convective heat transfer problems in porous media. The intrusive Proper Orthogonal Decomposition aided Reduced-Basis (POD-RB) technique is employed to reduce the porous medium formulation of the incompressible Reynolds-Averaged Navier–Stokes (RANS) equations coupled with heat transfer. Instead of resolving the exact flow configuration with high fidelity, the porous medium formulation solves a homogenized flow in which the fluid-structure interactions are captured via volumetric flow resistances with nonlinear, semi-empirical friction correlations. A supremizer approach is implemented for the stabilization of the reduced fluid dynamics equations. The reduced nonlinear flow resistances are treated using the Discrete Empirical Interpolation Method (DEIM), while the turbulent eddy viscosity and diffusivity are approximated by adopting a Radial Basis Function (RBF) interpolation-based approach. The proposed method is tested using a 2D numerical model of the Molten Salt Fast Reactor (MSFR), which involves the simulation of both clean and porous medium regions in the same domain. For the steady-state example, five model parameters are considered to be uncertain: the magnitude of the pumping force, the external coolant temperature, the heat transfer coefficient, the thermal expansion coefficient, and the Prandtl number. For transient scenarios, on the other hand, the coastdown-time of the pump is the only uncertain parameter. The results indicate that the POD-RB-ROMs are suitable for the reduction of similar problems. The relative L² errors are below 3.34% for every field of interest for all cases analyzed, while the speedup factors vary between 54 (transient) and 40,000 (steady-state). Full article

(This article belongs to the Special Issue Reduced Order Models for Computational Fluid Dynamics)

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24 pages, 4139 KiB

Open AccessArticle

Stochastic Galerkin Reduced Basis Methods for Parametrized Linear Convection–Diffusion–Reaction Equations

by Sebastian Ullmann, Christopher Müller and Jens Lang

Fluids 2021, 6(8), 263; https://0-doi-org.brum.beds.ac.uk/10.3390/fluids6080263 - 22 Jul 2021

Cited by 1 | Viewed by 1742

Abstract

We consider the estimation of parameter-dependent statistics of functional outputs of steady-state convection–diffusion–reaction equations with parametrized random and deterministic inputs in the framework of linear elliptic partial differential equations. For a given value of the deterministic parameter, a stochastic Galerkin finite element (SGFE) method can estimate the statistical moments of interest of a linear output at the cost of solving a single, large, block-structured linear system of equations. We propose a stochastic Galerkin reduced basis (SGRB) method as a means to lower the computational burden when statistical outputs are required for a large number of deterministic parameter queries. Our working assumption is that we have access to the computational resources necessary to set up such a reduced-order model for a spatial-stochastic weak formulation of the parameter-dependent model equations. In this scenario, the complexity of evaluating the SGRB model for a new value of the deterministic parameter only depends on the reduced dimension. To derive an SGRB model, we project the spatial-stochastic weak solution of a parameter-dependent SGFE model onto a reduced basis generated by a proper orthogonal decomposition (POD) of snapshots of SGFE solutions at representative values of the parameter. We propose residual-corrected estimates of the parameter-dependent expectation and variance of linear functional outputs and provide respective computable error bounds. We test the SGRB method numerically for a convection–diffusion–reaction problem, choosing the convective velocity as a deterministic parameter and the parametrized reactivity or diffusivity field as a random input. Compared to a standard reduced basis model embedded in a Monte Carlo sampling procedure, the SGRB model requires a similar number of reduced basis functions to meet a given tolerance requirement. However, only a single run of the SGRB model suffices to estimate a statistical output for a new deterministic parameter value, while the standard reduced basis model must be solved for each Monte Carlo sample. Full article

(This article belongs to the Special Issue Reduced Order Models for Computational Fluid Dynamics)

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