Advertisement

Acceleration Procedures for The Numerical Simulation of Compressible and Incompressible Viscous Flows

  • M. O. Bristeau
  • R. Glowinski
  • J. Periaux
Part of the International Centre for Mechanical Sciences book series (CISM, volume 300)

Abstract

Applying operator splitting methods to the numerical simulation of compressible or incompressible viscous flows leads to the solution of Stokes type linear problems and of nonlinear elliptic systems. Once discretized, these problems involve a large number of variables and therefore require efficient solution methods.

In this paper, we discuss the solution of the Stokes subproblems by iterative methods preconditioned by a well chosen boundary operator. The nonlinear subproblems (which are highly advective) are solved by an iterative method of GMRES type (cf. [1]). Numerical results corresponding to the simulation of flow around (and/or inside) air intakes and bodies illustrate the possibilities of the methods discussed here.

Keywords

Computational Fluid Dynamics Conjugate Gradient Boundary Node Stokes Problem Conjugate Gradient Algorithm 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    P. N. Brown, Y. Saad, Hybrid Krylov Methods for Nonlinear Systems of equations, Lawrence Livermore National Laboratory Research Report UCLR-97645, Nov. 1987.Google Scholar
  2. [2]
    R. Glowinski, Numerical Methods for Nonlinear Variational Problems, Springer-Verlag, New York, 1984.CrossRefMATHGoogle Scholar
  3. [3]
    M. O. Bristeau, R. Glowinski, J. Pénaux, Numerical Methods for the Navier-Stokes Equations. Application to the Simulation of Compressible and Incompressible Viscous Flows, Computer Physics Reports, 6, (1987), pp. 73–187.CrossRefGoogle Scholar
  4. [4]
    R. Glowinski, O. Pironneau, On numerical methods for the Stokes problem, Chapter 13 of Energy Methods in Finite Element Analysis, R. Glowinski, E. Y. Rodin, O. C. Zienkiewicz eds., J. Wiley, Chichester, 1979, pp. 243–264.Google Scholar
  5. [5]
    C. Bègue, R. Glowinski, J. Périaux, Détermination d’un opérateur de préconditionnement pour la résolution itérative du problème de Stokes dans la formulation d’Helmholtz, C. R. Acad. Sciences, Paris, T.306, Série I, pp. 247–252, 1988.Google Scholar
  6. [6]
    R. Glowinski, On a new preconditioner for the Stokes problem, MatematicaAplicada e Computational, 6, (1987), 2, pp. 123–140.MathSciNetMATHGoogle Scholar
  7. [7]
    N. Goutal, Résolution des équations de Saint-Venant, Ph.D. dissertation, University of Paris VI, Paris, France, Feb. 1987.Google Scholar
  8. [8]
    C. Bègue, M. O. Bristeau, R. G. Glowinski, B. Mantel, J. Périaux, G. Rogé, Sur l’Analyse de Fourier d’un opérateur de préconditionnement pour le problème de Stokes généralisé des écoulements visqueux compressibles (to appear).Google Scholar
  9. [9]
    M. O. Bristeau, O. Pironneau, R. Glowinski, J. Périaux, P. Perrier and G. Poirier, On the numerical solution of nonlinear problems in Fluid Dynamics by least squares and finite element methods (II). Application to transonic flows simulations, Comp. Meth. in Appl. Mech. Eng., 51, (1985), pp. 363–394.CrossRefMATHGoogle Scholar
  10. [10]
    L. B. Wigton, N. J. Yu and D. P. Young, GMRES Acceleration of Computational FluidGoogle Scholar
  11. Dynamics Codes, AIAA 7th Computational Fluid Dynamics Conference, Cincinnati, Ohio July, 1985, Paper 85–1494, pp. 67–74.Google Scholar
  12. [11]
    C. Bègue, M. O. Bristeau, R. Glowinski, B. Mantel, J. Pénaux, Acceleration of the convergence for viscous flow calculations, in Numets 87, Vol, 2, C. N. Pande, J. Middleton eds., Martins Nighoff Publishers, Dordrecht, 1987, pp. T4/1 - T4/20.Google Scholar
  13. [12]
    M. Mallet, J. Périaux, B. Stoufflet, On fast Euler and Navier-Stokes solvers, Proceedings of Me 7M GAMM Conference on Numerical Methods in Fluid Mechanics, Louvain, Belgium, 1987.Google Scholar
  14. [13]
    J. E. Dennis, R. B. Schnabel, Numerical Methods for Unconstrained Optimization and Nonlinear Equations, Prentice Hall, Englewood Cliffs, N. J. 1983.Google Scholar
  15. [14]
    C. Rogé, Ph.D. dissertation, University of Paris VI, Paris, France, 1988/1989 (to appear).Google Scholar
  16. [15]
    M. O. Bristeau, R Glowinski, B. Mantel, J. Pénaux, C. Rogé, Self-adaptive finite element method for 3D compressible Navier-Stokes flow simulation in Aerospace Engineering, Proceedings of the 11th Int. Conf. on Num. Meth. in Fluid Dynamics,Williamsburg, USA, June 1988 (to appear).Google Scholar
  17. [16]
    J. Cahouet, J. P. Chabard, Multi-domains and multi-solvers finite element approach for the Stokes problem, in Innovative Numerical Methods in Engineering, R. P. Shaw et al, eds., Springer-Verlag, Berlin, 1986, pp. 317–322.Google Scholar
  18. [17]
    V. Girault, P. A. Raviart, Finite Element Methods for Navier-Stokes Equations, Springer-Verlag, Berlin, 1986.CrossRefMATHGoogle Scholar
  19. [18]
    D. N. Arnold, F. Brersi, M. Fortin, A stable finite element for the Stokes equations, Calcolo, 21, (1984), 337.MathSciNetCrossRefMATHGoogle Scholar
  20. [19]
    M. O. Bristeau, R. Glowinski, B. Mantel, J. Périaux, G. Roge, Acceleration of compressible Navier-Stokes calculations, to appear in the Proceedings of the IMA International Conference on Computational Fluid Dynamics, Oxford, 1988.Google Scholar

Copyright information

© Springer-Verlag Wien 1989

Authors and Affiliations

  • M. O. Bristeau
    • 1
  • R. Glowinski
    • 2
  • J. Periaux
    • 3
  1. 1.INRIALe ChesnayFrance
  2. 2.University of HoustonHoustonUSA
  3. 3.AMD/BASaint CloudFrance

Personalised recommendations