Advertisement

Journal of Scientific Computing

, Volume 73, Issue 1, pp 157–177 | Cite as

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

  • Giuseppe Pitton
  • Gianluigi Rozza
Article

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.

Keywords

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

Notes

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.

References

  1. 1.
    Abdulle, A., Budác, O.: A Petrov–Galerkin reduced basis approximation of the Stokes equation in parametrized geometries. C. R. Acad. Sci. Ser. I Math. 353(7), 641–645 (2015)CrossRefzbMATHGoogle Scholar
  2. 2.
    Ambrosetti, A., Prodi, G.: A Primer of Nonlinear Analysis. Cambridge University Press, Cambridge (1993)zbMATHGoogle Scholar
  3. 3.
    Binev, P., Cohen, A., Dahmen, W., DeVore, R., Petrova, G., Wojtaszczyk, P.: Convergence rates for greedy algorithms in reduced basis methods. SIAM J. Math. Anal. 43(3), 1457–1472 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Brezzi, F., Rappaz, J., Raviart, P.: Finite dimensional approximation of nonlinear problems. Part I: branches of nonsingular solutions. Numer. Math. 36(1), 1–25 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Brezzi, F., Rappaz, J., Raviart, P.: Finite dimensional approximation of nonlinear problems. Part II: limit points. Numer. Math. 37(1), 1–28 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Brezzi, F., Rappaz, J., Raviart, P.: Finite dimensional approximation of nonlinear problems. Part III: simple bifurcation points. Numer. Math. 38(1), 1–30 (1982)CrossRefzbMATHGoogle Scholar
  7. 7.
    Buffa, A., Maday, Y., Patera, A., Turinici, G.: A priori convergence of the greedy algorithm for the parameterized reduced basis. ESAIM Math. Model. Numer. Anal. 46(3), 595–603 (2011)CrossRefzbMATHGoogle Scholar
  8. 8.
    Caiazzo, A., Iliescu, T., Volker, J., Schyschlowa, S.: A numerical investigation of velocity-pressure reduced order models for incompressible flows. J. Comput. Phys. 259, 598–616 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Canuto, C., Hussaini, M., Quarteroni, A., Zhang, T.: Spectral Methods Fundamentals in Single Domains. Scientific Computation. Springer, Berlin (2006)zbMATHGoogle Scholar
  10. 10.
    Canuto, C., Hussaini, M., Quarteroni, A., Zhang, T.: Spectral Methods Evolution to Complex Geometries and Applications to Fluid Dynamics. Scientific Computation. Springer, Berlin (2007)zbMATHGoogle Scholar
  11. 11.
    Chandrasekhar, S.: Hydrodynamic and Hydromagnetic Stability. Dover, New York (1982)zbMATHGoogle Scholar
  12. 12.
    Cliffe, K., Hall, E., Houston, P., Phipps, E., Salinger, A.: Adaptivity and a posteriori error control for bifurcation problems. III: incompressible fluid flow in open systems with O(2) symmetry. J. Sci. Comput. 52(1), 153–179 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Dahmen, W.: How to best sample a solution manifold? In: Pfander, G. (ed.) Sampling Theory—A Renaissance, Applied and Numerical Harmonic Analysis. Springer, Birkhäuser (2015)Google Scholar
  14. 14.
    Deparis, S., Løvgren, A.E.: Stabilized reduced basis approximation of incompressible three-dimensional Navier–Stokes equations in parametrized deformed domains. J. Sci. Comput. 50(1), 198–212 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Deville, M., Fischer, P., Mund, E.: High-Order Methods for Incompressible Fluid Flow. Cambridge Monographs on Applied and Computational Mathematics. Cambridge University Press, Cambridge (2002)CrossRefzbMATHGoogle Scholar
  16. 16.
    Du, Q., Gunzburger, M.: Model reduction by proper orthogonal decomposition coupled with centroidal Voronoi tessellation. In: Proceedings of Fluids Engineering Division Summer Meeting, ASME, vol. 1 (2002)Google Scholar
  17. 17.
    Du, Q., Gunzburger, M.: Centroidal Voronoi Tessellation Based Proper Orthogonal Decomposition Analysis. Birkhauser, Basel (2003)CrossRefzbMATHGoogle Scholar
  18. 18.
    Fischer, P., Lottes, J., Kerkemeier, S.: Nek5000 web page. http://nek5000.mcs.anl.gov (2008)
  19. 19.
    Galdi, G.: Mathematics of complexity and dynamical systems. In: Meyers, R. (ed.) Navier–Stokes Equations: A Mathematical Analysis, pp. 1009–1042. Springer, Berlin (2011)Google Scholar
  20. 20.
    Gelfgat, A., Bar-Yoseph, P., Yarin, A.: Stability of multiple steady states of convection in laterally heated cavities. J. Fluid Mech. 388, 315–334 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Golub, G., Loan, C.V.: Matrix Computations. Johns Hopkins University Press, Baltimore (2012)zbMATHGoogle Scholar
  22. 22.
    Gräbner, N., Mehrmann, V., Quraishi, S., Schröder, C., von Wagner, U.: Numerical methods for parametric model reduction in the simulation of disk brake squeal. ZAMM J. Appl. Math. Mech. Z. Angew. Math. Mech. (2016). doi: 10.1002/zamm.201500217
  23. 23.
    Haasdonk, B., Ohlberger, M.: Reduced basis method for finite volume approximations of parametrized linear evolution equations. ESAIM Math. Model. Numer. Anal. 42(2), 277–302 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Herrero, H., Maday, Y., Pla, F.: RB (reduced basis) for RB (Rayleigh–Bénard). Comput. Methods Appl. Mech. Eng. 261–262, 132–141 (2013)CrossRefzbMATHGoogle Scholar
  25. 25.
    Holmes, P., Lumley, J., Berkooz, G., Rowley, C.: Turbulence, Coherent Structures, Dynamical Systems and Symmetry. Cambridge University Press, Cambridge (2012)CrossRefzbMATHGoogle Scholar
  26. 26.
    Lassila, T., Manzoni, A., Quarteroni, A., Rozza, G.: Model order reduction in fluid dynamics: challenges and perspectives. In: Quarteroni, A., Rozza, G. (eds.) Reduced Order Methods for Modeling and Computational Reduction, Modeling, Simulation and Applications, vol 9, chap 9, pp. 235–273. Springer, Milano (2014)Google Scholar
  27. 27.
    Løvgren, A., Maday, Y., Rønquist, E.: A reduced basis element method for the steady Stokes problem. ESAIM Math. Model. Numer. Anal. 40(3), 529–552 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Machiels, L., Maday, Y., Oliveira, I.B., Patera, A.T., Rovas, D.V.: Output bounds for reduced-basis approximations of symmetric positive definite eigenvalue problems. C. R. Acad. Sci. Ser. I Math. 331(2), 153–158 (2000)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Marsden, J., Hughes, T.: Mathematical Foundations of Elasticity. Dover Civil and Mechanical Engineering. Dover, New York (1994)Google Scholar
  30. 30.
    Nguyen, N., Rozza, G., Patera, A.: Reduced basis approximation and a posteriori error estimation for the time-dependent viscous Burgers’ equation. Calcolo 46, 157–185 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Noor, A., Peters, J.: Multiple-parameter reduced basis technique for bifurcation and post-buckling analyses of composite materials. Int. J. Numer. Methods Eng. 19, 1783–1803 (1983)CrossRefzbMATHGoogle Scholar
  32. 32.
    Pitton, G., Quaini, A., Rozza, G.: Computational reduction strategies for the detection of steady bifurcations in incompressible fluid dynamics: applications to Coanda effect. Report SISSA 35/2016/MATE (2016) (submitted) Google Scholar
  33. 33.
    Quaini, A., Glowinski, R., Čanić, S.: Symmetry breaking and Hopf bifurcation for incompressible viscous flow in a contraction-expansion channel. Int. J. Comput. Fluid Dyn. 30(1), 7–19 (2016)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Quarteroni, A., Valli, A.: Domain Decomposition Methods for Partial Differential Equations. Numerical Mathematics and Scientific Computation. Oxford University Press, Oxford (1999)zbMATHGoogle Scholar
  35. 35.
    Roux, B. (ed.): Numerical Simulation of Oscillatory Convection in Low-Pr Fluids, Notes on Numerical Fluid Mechanics and Multidisciplinary Design, vol. 27. Springer, Berlin (1990)Google Scholar
  36. 36.
    Rovas, D.: Reduced-basis output bound methods for parametrized partial differential equations. Ph.D. thesis, Massachusetts Institute of Technology (2003)Google Scholar
  37. 37.
    Rozza, G., Veroy, K.: On the stability of the reduced basis method for Stokes equations on parametrized domains. Comput. Methods Appl. Mech. Eng. 196, 1244–1260 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Rozza, G., Huynh, D., Patera, A.: Reduced basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive partial differential equations. Arch. Comput. Methods Eng. 15(3), 229–275 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Rozza, G., Huynh, D., Manzoni, A.: Reduced basis approximation and a posteriori error estimation for Stokes flows in parametrized geometries: roles of the inf–sup stability constants. Numer. Math. 125(1), 115–152 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Ruelle, D., Takens, F.: On the nature of turbulence. Commun. Math. Phys. 20(3), 167–192 (1971)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Temlyakov, V.: Greedy approximation. Acta Numer. 17, 235–409 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Terragni, F., Vega, J.: On the use of POD-based ROMs to analyze bifurcations in some dissipative systems. Phys. D Nonlinear Phenom. 241(17), 1393–1405 (2012)CrossRefzbMATHGoogle Scholar
  43. 43.
    Timoshenko, S., Gere, J.: Theory of Elastic Stability. Dover Civil and Mechanical Engineering. Dover, New York (2009)Google Scholar
  44. 44.
    Tomboulides, A., Lee, J., Orszag, S.: Numerical simulation of low Mach number reactive flows. J. Sci. Comput. 12(2), 139–167 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Volkwein, S.: Proper Orthogonal Decomposition: Theory and Reduced-Order Modelling. University of Konstanz (2013)Google Scholar
  46. 46.
    Yano, M., Patera, A.T.: A space–time variational approach to hydrodynamic stability theory. Proc. R. Soc. A 496(2155), 20130036 (2013)Google Scholar

Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.mathLab, Mathematics AreaSISSA, International School for Advanced StudiesTriesteItaly

Personalised recommendations