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Hi-POD Solution of Parametrized Fluid Dynamics Problems: Preliminary Results

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Model Reduction of Parametrized Systems

Part of the book series: MS&A ((MS&A,volume 17))

Abstract

Numerical modeling of fluids in pipes or network of pipes (like in the circulatory system) has been recently faced with new methods that exploit the specific nature of the dynamics, so that a one dimensional axial mainstream is enriched by local secondary transverse components (Ern et al., Numerical Mathematics and Advanced Applications, pp 703–710. Springer, Heidelberg, 2008; Perotto et al., Multiscale Model Simul 8(4):1102–1127, 2010; Perotto and Veneziani, J Sci Comput 60(3):505–536, 2014). These methods—under the name of Hierarchical Model (Hi-Mod) reduction—construct a solution as a finite element axial discretization, completed by a spectral approximation of the transverse dynamics. It has been demonstrated that Hi-Mod reduction significantly accelerates the computations without compromising the accuracy. In view of variational data assimilation procedures (or, more in general, control problems), it is crucial to have efficient model reduction techniques to rapidly solve, for instance, a parametrized problem for several choices of the parameters of interest. In this work, we present some preliminary results merging Hi-Mod techniques with a classical Proper Orthogonal Decomposition (POD) strategy. We name this new approach as Hi-POD model reduction. We demonstrate the efficiency and the reliability of Hi-POD on multiparameter advection-diffusion-reaction problems as well as on the incompressible Navier-Stokes equations, both in a steady and in an unsteady setting.

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Notes

  1. 1.

    All the experiments have been performed using MATLAB®; R2010a 64-bit on a Fujitsu Lifebook T902 equipped with a 2.70 GHz i5 (3rd generation) vPro processor and 8 GB of RAM.

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Acknowledgements

The third and the fifth author gratefully acknowledge the NSF project DMS 1419060 “Hierarchical Model Reduction Techniques for Incompressible Fluid-Dynamics and Fluid-Structure Interaction Problems” (P.I. Alessandro Veneziani). The third author acknowledges also the financial support of GNCS-Project “Tecniche di riduzione della complessità computazionale per le scienze applicate”.

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Authors and Affiliations

  1. MOX, Dipartimento di Matematica, Politecnico di Milano, Piazza Leonardo da Vinci 32, I-20133, Milano, Italy

    Davide Baroli & Simona Perotto

  2. Faculté des Sciences, de la Technologie et de la Communication, Université du Luxembourg, 6, rue Richard Coudenhove-Kalergi, L-1359, Luxembourg, Luxembourg

    Davide Baroli

  3. Politecnico di Milano, Piazza Leonardo da Vinci 32, I-20133, Milano, Italy

    Cristina Maria Cova & Lorenzo Sala

  4. Université de Strasbourg, Laboratoire IRMA, 7, rue René Descartes, 67084, Strasbourg Cedex, France

    Lorenzo Sala

  5. Department of Mathematics and Computer Science, Emory University, Atlanta, GA, 30322, USA

    Alessandro Veneziani

Authors
  1. Davide Baroli
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  2. Cristina Maria Cova
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  3. Simona Perotto
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  4. Lorenzo Sala
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  5. Alessandro Veneziani
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Corresponding author

Correspondence to Simona Perotto .

Editor information

Editors and Affiliations

  1. Computational Methods in Systems and Control Theory, Max Planck Institute for Dynamics of Complex Technical Systems, Magdeburg, Germany

    Peter Benner

  2. Institute for Computational and Applied Mathematics, University of Münster , Münster, Germany

    Mario Ohlberger

  3. Massachusetts Institute of Technology, MIT , Cambridge, Massachusetts, USA

    Anthony Patera

  4. SISSA MathLab, International School for Adv. Studies , Trieste, Italy

    Gianluigi Rozza

  5. Institute of Numerical Mathematics, Ulm University , Ulm, Germany

    Karsten Urban

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Baroli, D., Cova, C.M., Perotto, S., Sala, L., Veneziani, A. (2017). Hi-POD Solution of Parametrized Fluid Dynamics Problems: Preliminary Results. In: Benner, P., Ohlberger, M., Patera, A., Rozza, G., Urban, K. (eds) Model Reduction of Parametrized Systems. MS&A, vol 17. Springer, Cham. https://doi.org/10.1007/978-3-319-58786-8_15

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