Model Reduction on Inertial Manifolds of Navier–Stokes Equations Through Multi-scale Finite Element

Conference paper


Multilevel finite element method is used to approach the Approximate Inertial Manifolds (AIMs) from viewpoint of nonlinear dynamics, in the computational fluid dynamics. By this method, an unknown variable is divided into two components, namely, the large eddy and small eddy components. With the introduction of an AIMs, the interaction between large eddy and small eddy components, which is negligible if standard Galerkin algorithm is used to approach the original governing equations, is considered essentially, and consequently a coarse grid finite element space and a fine grid incremental finite element space are introduced to approach the two components. By this method, the flow field of incompressible flows around airfoil is simulated numerically as an example, and velocity and pressure distributions of the flow field are obtained accurately. The results show that there exists less degrees-of-freedom in the discretized system in comparison with the traditional methods, and large computing time can be saved by this efficient method. The small eddy component can be captured by AIMs, and an accurate result can also be obtained.


Model Reduction Global Attractor Quadrangle Element Finite Element Space Inertial Manifold 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



This research is supported by Program for New Century Excellent Talents in University in China, No. NCET-07-0685.


  1. 1.
    Steindl A, Troger H (2001) Methods for dimension reduction and their application in nonlinear dynamics. Int J Solids Struct 38:2131–2147MATHCrossRefGoogle Scholar
  2. 2.
    Zhang JZ, Liu Y, Chen DM (2005) Error estimate for influence of model reduction of nonlinear dissipative autonomous dynamical system on long-term behaviours. Appl Math Mech 26: 938–943MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Guckenheimer J, Holmes P (1983) Nonlinear oscillations, dynamical system, and bifurcations of vector fields. Springer, New YorkGoogle Scholar
  4. 4.
    Wiggins S (1990) Introduction to applied nonlinear dynamical systems and chaos. Springer, New YorkMATHCrossRefGoogle Scholar
  5. 5.
    Seydel R (1994) Practical bifurcation and stability analysis: from equilibrium to chaos. Springer, New YorkMATHGoogle Scholar
  6. 6.
    Friswell MI, Penny JET, Garvey SD (1996) The application of the IRS and balanced realization methods to obtain reduced models of structures with local nonlinearities. J Sound Vib 196: 453–468CrossRefGoogle Scholar
  7. 7.
    Fey RHB, van Campen DH, de Kraker A (1996) Long term structural dynamics of mechanical system with local nonlinearities. ASME J Vib Acoust 118:147–163CrossRefGoogle Scholar
  8. 8.
    Kordt M, Lusebrink H (2001) Nonlinear order reduction of structural dynamic aircraft models. Aerosp Sci Technol 5:55–68MATHCrossRefGoogle Scholar
  9. 9.
    Slaats PMA, de Jongh J, Sauren AAHJ (1995) Model reduction tools for nonlinear structural dynamics. Comput Struct 54:1155–1171MATHCrossRefGoogle Scholar
  10. 10.
    Zhang JZ (2001) Calculation and bifurcation of fluid film with cavitation based on variational inequality. Int J Bifurcat Chaos 11:43–55CrossRefGoogle Scholar
  11. 11.
    Murota K, Ikeda K (2002) Imperfect bifurcation in structures and materials. Springer, New YorkMATHGoogle Scholar
  12. 12.
    Marion M, Temam R (1989) Nonlinear Galerkin methods. Siam J Numer Anal 26:1139–1157MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Temam R (1997) Infinite-dimensional dynamical system in mechanics and physics. Springer, New YorkGoogle Scholar
  14. 14.
    Titi ES (1990) On approximate inertial manifolds to the Navier–Stokes equations. J Math Anal Appl 149:540–557MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Jauberteau F, Rosier C, Temam R (1990) A nonlinear Galerkin method for the Navier–Stokes equations. Comput Methods Appl Mech Eng 80:245–260MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Schmidtmann O (1996) Modelling of the interaction of lower and higher modes in two-dimensional MHD-equations. Nonlinear Anal Theory Methods Appl 26:41–54MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Chueshov ID (1996) On a construction of approximate inertial manifolds for second order in time evolution equations. Nonlinear Anal Theory Methods Appl 26:1007–1021MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Rezounenko AV (2002) Inertial manifolds for retarded second order in time evolution equations. Nonlinear Anal 51:1045–1054MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Laing CR, McRobie A, Thompson JMT (1999) The post-processed Galerkin method applied to non-linear shell vibrations. Dyn Stab Syst 14:163–181MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Zhang JZ, van Campen DH, Zhang GQ, Bouwman V, ter Weeme JW (2001) Dynamic stability of doubly curved orthotropic shallow shells under impact. AIAA J 39:956–961CrossRefGoogle Scholar
  21. 21.
    Foias C, Sell GR, Temam R (1985) Varietes Inertielles des Equations Differentielles Dissipatives. C R Acad Sci Paris Ser I Math 301:139–141MathSciNetMATHGoogle Scholar
  22. 22.
    Chow SN, Lu K (2001) Invariant manifolds for flows in Banach space. J Differ Equ 74:285–317MathSciNetCrossRefGoogle Scholar
  23. 23.
    Foias C, Manley O, Temam R (1987) On the interaction of small and large eddies in turbulent flows. C R Acad Sci Paris Ser I Math 305:497–500MathSciNetMATHGoogle Scholar
  24. 24.
    Foias C, Sell GR, Titi ES (1989) Exponential tracking and approximation of inertial manifolds for dissipative nonlinear equations. J Dyn Differ Equ 1:199–244MathSciNetMATHCrossRefGoogle Scholar
  25. 25.
    Rezounenko AV (2004) Investigations of retarded PDEs of second order in time using the method of inertial manifolds with delay. Ann Inst Fourier (Grenoble) 54:1547–1564MathSciNetMATHCrossRefGoogle Scholar
  26. 26.
    Zhang JZ, Liu Y, Cheng DM (2005) Error estimate for the influence of model reduction of nonlinear dissipative autonomous dynamical system on the long-term behaviors. J Appl Math Mech 26(7):938–943MATHCrossRefGoogle Scholar
  27. 27.
    Zhang JZ, Liu Y, Lei PF, Sun X (2007) Dynamic snap-through buckling analysis of shallow arches under impact load based on approximate inertial manifolds. Dyn Continuous Discrete Impulsive Syst Ser B (DCDIS-B) 14:287–291Google Scholar
  28. 28.
    He YN, Li KT (1999) Optimum finite element nonlinear Galerkin algorithm for the Navier–Stokes equations. Math Numer Sin 21(1):29–38 (in Chinese)MathSciNetMATHGoogle Scholar
  29. 29.
    Ren X, Li BH, Yin XY, Gao G (2007) Calculation of airfoil flows using GAO-YONG turbulence equations. J Aerosp Power 22(1):73–78 (in Chinese)Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.School of Energy and Power EngineeringXi’an Jiaotong UniversityXi’anPeople’s Republic of China

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