Adaptive finite element methods for three dimensional compressible viscous flow simulation in aerospace engineering

  • M. O. Bristeau
  • R. Glowinski
  • B. Mantel
  • J. Périaux
  • G. Rogé
Contributed Papers
Part of the Lecture Notes in Physics book series (LNP, volume 323)


We have briefly discussed here the numerical solution of the compressible Navier-Stokes equations written in non conservative form. We have shown by these first results that approximations satisfying some “Inf-sup” condition lead, for moderate Reynolds and Mach numbers, to accurate results without upwinding or artificial viscosity. The solution techniques which have been discussed are extended presently to conservative formulations in view of genuine hypersonic simulations including real gas effects. We can hope from the first results that, for higher Mach and Reynolds numbers, we will obtain accurate solutions while introducing less dissipation in the schemes; the accuracy, particularly of pressure, being a critical point in view of further calculations in hypersonic design.


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  1. [1]
    R. Peyret and T.D. Taylor, Computational Methods for Fluid Flow (Springer-Verlag, New-York, 1982).Google Scholar
  2. [2]
    M.O. Bristeau, R. Glowinski, B. Mantel, J. Périaux and G. Rogé, Acceleration of Compressible Navier-Stokes calculations, Proceedings of ICFD 88 Conference on Numerical Methods for Fluid Dynamics, Oxford, March 21–24,1988. K.W. Morton ed.Google Scholar
  3. [3]
    M.O. Bristeau, R. Glowinski and J. Périaux, Numerical Methods for the Navier-Stokes equations. Applications to the simulation of compressible and incompressible viscous flows. Computer Physics Report 6 (1987), North-Holland, Amsterdam, pp. 73–187.Google Scholar
  4. [4]
    J. Cahouet and J.P. Chabard, Multi-Domains and Multi-Solvers Finite Element Approach for the Stokes problem, in: Innovative Numerical Methods in Engineering, eds., R.P. Shaw, J. Périaux, A. Chadouet, J. Wu, C. Marino, C.A. Brebbia (Springer-Verlag, Berlin, 1986), p. 317–322.Google Scholar
  5. [5]
    C. Bègue, R. Glowinski and J. Périaux, Détermination d'un opérateur de préconditionnement pour la résolution itérative du problème de Stokes dans la formulation d'Helmholtz, C.R. Acad. Sci. Paris, t. 306, Série I, p. 247–252, 1988.Google Scholar
  6. [6]
    C. Bègue, M.O. Bristeau, R. Glowinski, B. Mantel, J. Périaux and G. Rogé, Accélération de la convergence dans le calcul des écoulements de fluides visqueux. lère partie: cas incompressible et cas compressible non conservatif. Rapport INRIA n° 861, 1988.Google Scholar
  7. [7]
    G. Rogé, Approximation mixte et accélération de la convergence pour la résolution des équations de Navier-Stokes compressible en éléments finis, Thèse de 3ème cycle, Université Pierre et Marie Curie, Paris 6, à paraître.Google Scholar
  8. [8]
    R. Glowinski, Numerical Methods for Nonlinear Variational Problems (Springer-Verlag, New-York, 1984).Google Scholar
  9. [9]
    Y. Saad and M.H. Schultz, GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems, SIAM J. Sci. Stat. Comp., 7 (1986), pp. 856–869.Google Scholar
  10. [10]
    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
  11. [11]
    V. Girault and P.A. Raviart, Finite Element Methods for Navier-Stokes Equations (Springer-Verlag, Berlin, 1986).Google Scholar
  12. [12]
    D.N. Arnold, F. Brezzi and M. Fortin, A stable finite element for the Stokes equations, Calcolo 21 (1984) 337.Google Scholar
  13. [13]
    J. Peraire, M. Vahdati, K. Morgan and O.C. Zienkiewicz, Adaptive remeshing for compressible flow computations, J. Comp. Phys. 72,2, 1987.Google Scholar
  14. [14]
    M.C. Rivara, Algorithms for refining triangular grids suitable for adaptive and multigrid techniques, Int. J. Num. Meth. in Eng., Vol. 20, pp. 745–756 (1984).Google Scholar
  15. [15]
    I. Babuska and W.C. Rheinbold, A posteriori error estimates for the Finite Element Method, Intern. J. Numer. Meth. Eng. 112 (1978) 1597.Google Scholar
  16. [16]
    R.E. Bank, Analysis of a local a posteriori error estimate for elliptic equations in Accuracy Estimates and Adaptivity for Finite Elements, J. Wiley & Sons, New-York, 1986, pp. 119–128.Google Scholar
  17. [17]
    Y. Secretan, G. Dhatt, D. Nguyen, Compressible Viscous Flow around a NACA0012 airfoil, Numerical Simulation of Compressible Navier-Stokes flows, Notes on Num. Fluid Mech., Vieweg, Vol. 18, 1987.Google Scholar

Copyright information

© Springer-Verlag 1989

Authors and Affiliations

  • M. O. Bristeau
    • 1
  • R. Glowinski
    • 2
  • B. Mantel
    • 3
  • J. Périaux
    • 3
  • G. Rogé
    • 3
  1. 1.INRIALe Chesnay CedexFrance
  2. 2.INRIAUniversity of HoustonHoustonUSA
  3. 3.AMD/BASt Cloud CedexFrance

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