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On the Application of Reduced Basis Methods to Bifurcation Problems in Incompressible Fluid Dynamics

Abstract

In this paper we apply a reduced basis framework for the computation of flow bifurcation (and stability) problems in fluid dynamics. The proposed method aims at reducing the complexity and the computational time required for the construction of bifurcation and stability diagrams. The method is quite general since it can in principle be specialized to a wide class of nonlinear problems, but in this work we focus on an application in incompressible fluid dynamics at low Reynolds numbers. The validation of the reduced order model with the full order computation for a benchmark cavity flow problem is promising.

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Notes

  1. 1.

    The notation chosen for the function spaces may need clarification. We refer with \(H^1_0\) to the Sobolev space with zero trace at the boundary (velocity), and with \(L^2_0\) to the Lebesgue \(L^2\) functions with zero average (pressure).

  2. 2.

    We refer the interested reader to [38] for a brief overview on the history of the Reduced Basis Method and a review of many sampling techniques.

  3. 3.

    Which is usually dependent on the discretization parameter such as grid size.

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Acknowledgements

The authors acknowledge Dr. E. Merzari for his help with the Nek5000 software and for the useful discussions, and the Nek5000 community in general, Dr. F. Ballarin for the insights on approximation stability. G. Pitton has been supported by the pre-doc program at SISSA. G. Rozza acknowledges the support of NOFYSAS Excellence Grant Program at SISSA and INDAM-GNCS Activity Group (2015 and 2016 projects), as well as European Union Funding for Research and Innovation—Horizon 2020 Program—in the framework of European Research Council Executive Agency: H2020 ERC CoG 2015 AROMA-CFD Project 681447 “Advanced Reduced Order Methods with Applications in Computational Fluid Dynamics”. The motivation for developing this work came from Prof. A.T. Patera (MIT) and from Prof. J. Rappaz (EPFL). We acknowledge Prof. F. Brezzi (IUSS, Pavia) for insights and some references. We gratefully thank Prof. A. Quaini for ongoing collaboration on this topic with the Mathematics Department at University of Houston, USA. The computing resources have been provided by the Sis14_COGESTRA cpu time grant allocation at CINECA, Bologna, Italy.

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Correspondence to Gianluigi Rozza.

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Pitton, G., Rozza, G. On the Application of Reduced Basis Methods to Bifurcation Problems in Incompressible Fluid Dynamics. J Sci Comput 73, 157–177 (2017). https://doi.org/10.1007/s10915-017-0419-6

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Keywords

  • Reduced basis method
  • Proper orthogonal decomposition
  • Steady bifurcation
  • Hopf bifurcation
  • Navier–Stokes
  • Flow stability
  • Spectral element method