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Model Order Reduction in Fluid Dynamics: Challenges and Perspectives

  • Chapter
Reduced Order Methods for Modeling and Computational Reduction

Part of the book series: MS&A - Modeling, Simulation and Applications ((MS&A,volume 9))

Abstract

This chapter reviews techniques of model reduction of fluid dynamics systems. Fluid systems are known to be difficult to reduce efficiently due to several reasons. First of all, they exhibit strong nonlinearities — which are mainly related either to nonlinear convection terms and/or some geometric variability — that often cannot be treated by simple linearization. Additional difficulties arise when attempting model reduction of unsteady flows, especially when long-term transient behavior needs to be accurately predicted using reduced order models and more complex features, such as turbulence or multiphysics phenomena, have to be taken into consideration. We first discuss some general principles that apply to many parametric model order reduction problems, then we apply them on steady and unsteady viscous flows modelled by the incompressible Navier-Stokes equations. We address questions of inf-sup stability, certification through error estimation, computational issues and — in the unsteady case — long-time stability of the reduced model. Moreover, we provide an extensive list of literature references.

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Notes

  1. 1.

    1 In practice, N r POD modes are required to resolve the first N r /2 temporal harmonics, and these can be computed from N s = 2N r snapshots [96].

  2. 2.

    2 For numerical stability reasons the PODeigenvalues are usually not computed from the correlation matrix itself, but rather as the squares of the singular values of the snapshot matrix obtained by collecting all the snapshots as column vectors.

  3. 3.

    3 Gram-Schmidt orthonormalization is required in order to ensure the algebraic stability of the reduced basis approximation. Furthermore, in case of parameter-dependent geometries, the velocity space has to be enriched, as detailed in Sect. 9.3.

  4. 4.

    4 This algorithm has been first introduced in [64] for both coercive and noncoercive problems, analyzed in [107] in the coercive case and afterwards improved in [32]. A general version using the so-called “natural norm” [110] has been analyzed in [61], where it has been applied to noncoercive problems such as Helmholtz equations — the simpler coercive case can be seen as a particular instance where the stability factor is just the coercivity constant.

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Authors and Affiliations

  1. MATHICSE-CMCS Modelling and Scientific Computing, Ecole Polytechnique Fédérale de Lausanne, Station 8, CH-1015, Lausanne, Switzerland

    Toni Lassila & Alfio Quarteroni

  2. SISSA Mathlab — International School for Advanced Studies, Santorio A, Via Bonomea 265, I-34136, Trieste, Italy

    Andrea Manzoni & Gianluigi Rozza

  3. MOX — Dipartimento di Matematica, Politecnico di Milano, P.za Leonardo da Vinci 32, I-20133, Milano, Italy

    Alfio Quarteroni

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  1. Toni Lassila
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  2. Andrea Manzoni
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  3. Alfio Quarteroni
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  4. Gianluigi Rozza
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  1. CMCS-MATHICSE, Ecole Polytechnique Fédérale de Lausanne, Lausanne, Switzerland

    Alfio Quarteroni

  2. MOX, Department of Mathematics “F. Brioschi”, Politecnico di Milano, Milan, Italy

    Alfio Quarteroni

  3. SISSA mathLab, International School for Advanced Studies, Trieste, Italy

    Gianluigi Rozza

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Lassila, T., Manzoni, A., Quarteroni, A., Rozza, G. (2014). Model Order Reduction in Fluid Dynamics: Challenges and Perspectives. In: Quarteroni, A., Rozza, G. (eds) Reduced Order Methods for Modeling and Computational Reduction. MS&A - Modeling, Simulation and Applications, vol 9. Springer, Cham. https://doi.org/10.1007/978-3-319-02090-7_9

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