Fas multigrid employing ILU/SIP smoothing: A robust fast solver for 3D transonic potential flow

  • A. J. van der Wees
Conference paper
Part of the Lecture Notes in Mathematics book series (LNM, volume 1228)


ILU/SIP is shown to be a very efficient and robust smoothing algorithm within the multigrid method for the solution of elliptic (subsonic) and mixed elliptic/hyperbolic (transonic) potential flow problems. The algorithm is fully implicit and fairly insensitive to large grid cell aspect ratios; in the hyperbolic regions of the flow the algorithm is uniformly stable.

It will also be shown that the best multigrid performance for 3D problems is obtained by performing a priori grid optimization, for which requirements will be derived. With an optimized grid, the method is fast for engineering applications and the physical quantities of interest are determined with great efficiency.


Multigrid Method Spanwise Direction Transonic Flow Grid Plane Hyperbolic Region 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

14. References

  1. [1]
    Boerstoel, J.W. and Kassies, A., Integrating multi-grid relaxation into a robust fast-solver for transonic potential flow around lifting aerofoils, AIAA Paper 83-1885, 1983.Google Scholar
  2. [2]
    Shmilovich, A. and Caughey, D.A., Application of the multi-grid method to calculations of transonic potential flow about wing-fuselage combinations, J. Comp. Phys. 48, pp. 462–484, 1982.CrossRefzbMATHGoogle Scholar
  3. [3]
    Jameson, A., Acceleration of transonic potential flow calculations on arbitrary meshes by the multiple grid method, AIAA Paper 79-1458 CP, 1979.Google Scholar
  4. [4]
    Schmidt, W. and Jameson, A., Applications of multi-grid methods for transonic flow calculations, Lecture Notes in Mathematics 960, Multi-grid Methods, Proceedings Köln-Porz, 1981. Edited by W. Hackbush and U. Trottenberg, Springer-Verlag.Google Scholar
  5. [5]
    Catherall, D., Optimum approximate factorisation schemes for 2D steady potential flows, AIAA Paper 81-1018-CP, 1981.Google Scholar
  6. [6]
    Holst, T., Numerical solution of transonic wing flow fields, AIAA Paper 82-0105, 1982.Google Scholar
  7. [7]
    South jr, J.C. and Hafez, M.M., Stability analysis of intermediate boundary conditions in approximate factorization schemes, AIAA Paper 83-1898-CP, 1983.Google Scholar
  8. [8]
    Brédif, M., Finite element calculation of potential flow around wings, ONERA-TP-1984-068, 1984.Google Scholar
  9. [9]
    Sankar, N.L., A multi-grid strongly implicit procedure for two-dimensional transonic potential flow problems, AIAA Paper 82-0931, 1982.Google Scholar
  10. [10]
    Van der Wees, A.J., van der Vooren, J. and Meelker, J.H., Robust calculation of 3D transonic potential flow based on the non-linear FAS multi-grid method and incomplete LU decomposition, AIAA Paper 83-1950-CP, 1983.Google Scholar
  11. [11]
    Van der Wees, A.J., Robust calculation of 3D transonic potential flow based on the non-linear FAS multi-grid method and a mixed ILU/SIP algorithm, Colloquium Topics in Numerical Analysis, J.G. Verwer (ed.), CWI Syllabus 5, 1985.Google Scholar
  12. [12]
    Jameson, A. and Caughey, D.A., A finite volume method for transonic potential flow calculations, AIAA Paper 77-635-CP, 1977.Google Scholar
  13. [13]
    Osher, S., Hafez, M.M. and Whithlow jr., W., Entropy condition satisfying approximations for the full potential equation of transonic flow, Math. of Comp., Vol. 44, Nr. 169, 1985.Google Scholar
  14. [14]
    Stüben, K. and Trottenberg, U., Multigrid methods: fundamental algorithms, model problem analysis and applications, Lecture Notes in Mathematics, see [3].Google Scholar
  15. [15]
    Brandt, A., Multi-level adaptive solutions to boundary value problems, Math. of Comp., Vol. 31, Nr. 138, 1977.Google Scholar
  16. [16]
    Meyerink, J.A. and van der Vorst, H.A., An iterative solution method for linear problems of which the coefficient matrix is a symmetric M-matrix, Math. of Comp., Vol. 31, Nr. 137, 1977.Google Scholar
  17. [17]
    Stone, H.L., Iterative solution of implicit approximations of multi-dimensional partial difference equations, SIAM J. Numer. Anal., Vol. 5, Nr. 3, 1968.Google Scholar
  18. [18]
    Zedan, M. and Schneider, G.E., 3-D Modified strongly implicit procedure for finite difference heat conduction modelling, AIAA Paper 81-1136, 1981.Google Scholar
  19. [19]
    Wesseling, P., A robust and efficient multigrid method, Lecture Notes in Mathematics, see [3].Google Scholar
  20. [20]
    Kettler, R., Analysis and comparison of relaxation schemes in robust multigrid and preconditioned conjugate gradient methods, Lecture Notes in Mathematics, see [3].Google Scholar
  21. [21]
    Jameson, A., Numerical solution of the three dimensional transonic flow over a yawed wing, AIAA Paper presented at the 1st AIAA Comp. Fluid, Dyn. Conf., Palm Springs, Cal., July 19–20, pp. 18–26, 1973.Google Scholar

Copyright information

© Springer-Verlag 1986

Authors and Affiliations

  • A. J. van der Wees
    • 1
  1. 1.Informatics DivisionNational Aerospace Laboratory NLREmmeloordThe Netherlands

Personalised recommendations