Model Order Reduction in Fluid Dynamics: Challenges and Perspectives

Lassila, Toni; Manzoni, Andrea; Quarteroni, Alfio; Rozza, Gianluigi

doi:10.1007/978-3-319-02090-7_9

Cited by 161 publications

(193 citation statements)

References 134 publications

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“…At least two approaches in the construction stage of the RB can be pursued: greedy algorithms and POD [1]. In this paper, we consider the latter [11,26,35,36].…”

Section: A Pod-galerkin Rom For Parametrized Navier-stokes Equationsmentioning

confidence: 99%

“…For nonlinear PDEs, several issues need, however, to be faced in order to guarantee efficiency, accuracy, and reliability, also when using ROMs. These include the efficient exploration of the parameters space to build reduced basis (RB) spaces that, ideally, should: (i) have low dimension and also the capability to capture fine physical features; (ii) be stable also for noncoercive problems as in the case of saddle-point problems; and (iii) allow accurate and fast estimation of stability factors, as recently pointed out in [1,2].…”

Section: Introduction and Motivationsmentioning

confidence: 99%

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Supremizer stabilization of POD–Galerkin approximation of parametrized steady incompressible Navier–Stokes equations

Ballarin

Manzoni

Quarteroni

et al. 2014

Numerical Meth Engineering

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269

318

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In this work, we present a stable proper orthogonal decomposition-Galerkin approximation for parametrized steady incompressible Navier-Stokes equations with low Reynolds number. Supremizer solutions are added to the reduced velocity space in order to obtain a stable reduced-order system, considering in particular the fulfillment of an inf-sup condition. The stability analysis is first carried out from a theoretical standpoint and then confirmed by numerical tests performed on a parametrized two-dimensional backward-facing step flow. 1137 ological developments were provided in more recent years [15][16][17][18], including the application to new unconventional fields [19,20].In recent years, growing attention has been devoted to the combination of Galerkin strategies with POD [21-23] and to stabilization techniques both for POD [9,[24][25][26] and RB methods [27][28][29][30].The aim of this work is to exploit some features of the RB framework in its state-of-the-art formulation for the stable and accurate approximation of parametrized flows in a POD setting, in order to improve the performance of the latter. In particular, we aim at investigating possible sources of pressure instabilities, in order to avoid spurious pressure modes in the POD approximation of parametrized flows. In fact, it is very well known that the FE approximation of Navier-Stokes (NS) equations presents two main difficulties that can make their numerical approximations meaningless: (i) the indefinite nature of the system and (ii) the stability loss due to convection dominant regimes for high Reynolds numbers. The first issue can be cured by using suitable velocitypressure FE spaces (also known as inf-sup stable elements), satisfying a discrete version of the so-called Ladyzhenskaya-Babuška-Brezzi (LBB) condition (see Section 4). Instead, in order to overcome the velocity stability loss for convection dominant regimes, we can rely on FE stabilization techniques, such as the streamline upwind Petrov-Galerkin (SUPG) method. We underline that stabilization techniques can also be employed to overcome pressure instability, for instance, through the pressure stabilizing Petrov-Galerkin (PSPG) method. Concerning pressure instability, a possible way to cure it in the RB context is the enrichment of velocity spaces through the so-called supremizers.Here, we want to analyze this supremizer stabilization technique to exploit it within a POD context and to develop a stability analysis based on the introduction of an LBB inf-sup condition at the reduced level, too. In this way, POD could benefit from a previously developed robust framework for the RB stabilization of viscous flows that can also manage with a correct pressure recovery. In this work, we test such a method on nonlinear steady viscous flows modeled by NS equations and characterized by both physical and geometrical parametrization.In the offline stage of the resulting strategy, several NS truth solutions are computed, and a POD is performed to extract a low-dimensional representation of both velo...

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“…At least two approaches in the construction stage of the RB can be pursued: greedy algorithms and POD [1]. In this paper, we consider the latter [11,26,35,36].…”

Section: A Pod-galerkin Rom For Parametrized Navier-stokes Equationsmentioning

confidence: 99%

Section: Introduction and Motivationsmentioning

confidence: 99%

Supremizer stabilization of POD–Galerkin approximation of parametrized steady incompressible Navier–Stokes equations

Ballarin

Manzoni

Quarteroni

et al. 2014

Numerical Meth Engineering

Self Cite

269

318

View full text Add to dashboard Cite

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“…The reduced subspace X n is constructed from a set of (well-chosen) full-order solutions, usually by exploiting one of the following techniques [22,38]:…”

Section: Reduced Subspaces and Projection-based Romsmentioning

confidence: 99%

“…However, in both cases no indications about the sign of the error are provided by the error bound (13), so that a correction based on (22) …”

Section: Rem-3: Linear Regression Modelmentioning

confidence: 99%

Accurate Solution of Bayesian Inverse Uncertainty Quantification Problems Combining Reduced Basis Methods and Reduction Error Models

Manzoni¹,

Pagani²,

Lassila³

2016

SIAM/ASA J. Uncertainty Quantification

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ReuseUnless indicated otherwise, fulltext items are protected by copyright with all rights reserved. The copyright exception in section 29 of the Copyright, Designs and Patents Act 1988 allows the making of a single copy solely for the purpose of non-commercial research or private study within the limits of fair dealing. The publisher or other rights-holder may allow further reproduction and re-use of this version -refer to the White Rose Research Online record for this item. Where records identify the publisher as the copyright holder, users can verify any specific terms of use on the publisher's website. TakedownIf you consider content in White Rose Research Online to be in breach of UK law, please notify us by emailing eprints@whiterose.ac.uk including the URL of the record and the reason for the withdrawal request. Abstract Computational inverse problems related to partial differential equations (PDEs) often contain nuisance parameters that cannot be effectively identified but still need to be considered as part of the problem. The objective of this work is to show how to take advantage of a reduced order framework to speed up Bayesian inversion on the identifiable parameters of the system, while marginalizing away the (potentially large number of) nuisance parameters. The key ingredients are twofold. On the one hand, we rely on a reduced basis (RB) method, equipped with computable a posteriori error bounds, to speed up the solution of the forward problem. On the other hand, we develop suitable reduction error models (REMs) to quantify in an inexpensive way the error between the full-order and the reduced-order approximation of the forward problem, in order to gauge the effect of this error on the posterior distribution of the identifiable parameters. Numerical results dealing with inverse problems governed by elliptic PDEs in the case of both scalar parameters and parametric fields highlight the combined role played by RB accuracy and REM effectivity.

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“…In the past few years, due to their relevance in realistic applications, a lot of interest has been devoted to discretization reduction techniques for parametrized Partial Differential Equation (PDE) problems (e.g., [3][4][5][6]). These techniques aim to define a suitable reduced order model which can be solved with marginal computational costs for different values of the model parameters.…”

Section: Introductionmentioning

confidence: 99%

Reduced Numerical Approximation of Reduced Fluid-Structure Interaction Problems With Applications in Hemodynamics

Colciago

Deparis

2018

Front. Appl. Math. Stat.

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This paper deals with fast simulations of the hemodynamics in large arteries by considering a reduced model of the associated fluid-structure interaction problem, which in turn allows an additional reduction in terms of the numerical discretisation. The resulting method is both accurate and computationally cheap. This goal is achieved by means of two levels of reduction: first, we describe the model equations with a reduced mathematical formulation which allows to write the fluid-structure interaction problem as a Navier-Stokes system with non-standard boundary conditions; second, we employ numerical reduction techniques to further and drastically lower the computational costs. The non standard boundary condition is of a generalized Robin type, with a boundary mass and boundary stiffness terms accounting for the arterial wall compliance. The numerical reduction is obtained coupling two well-known techniques: the proper orthogonal decomposition and the reduced basis method, in particular the greedy algorithm. We start by reducing the numerical dimension of the problem at hand with a proper orthogonal decomposition and we measure the system energy with specific norms; this allows to take into account the different orders of magnitude of the state variables, the velocity and the pressure. Then, we introduce a strategy based on a greedy procedure which aims at enriching the reduced discretization space with low offline computational costs. As application, we consider a realistic hemodynamics problem with a perturbation in the boundary conditions and we show the good performances of the reduction techniques presented in the paper. The results obtained with the numerical reduction algorithm are compared with the one obtained by a standard finite element method.The gains obtained in term of CPU time are of three orders of magnitude.

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Model Order Reduction in Fluid Dynamics: Challenges and Perspectives

Cited by 161 publications

References 134 publications

Supremizer stabilization of POD–Galerkin approximation of parametrized steady incompressible Navier–Stokes equations

Supremizer stabilization of POD–Galerkin approximation of parametrized steady incompressible Navier–Stokes equations

Accurate Solution of Bayesian Inverse Uncertainty Quantification Problems Combining Reduced Basis Methods and Reduction Error Models

Reduced Numerical Approximation of Reduced Fluid-Structure Interaction Problems With Applications in Hemodynamics

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