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A shock indicator for adaptive transonic flow computations

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Summary

An adaptive finite element method for the calculation of transonic potential flows was developed. A residual based error indicator is complemented by a shock indicator. For a good shock resolution mesh refinement as well as moving nodes were needed. An analysis of the method and computational results are given.

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References

  1. Berger, H. (1989): A convergent finite element formulation for transonic flow. Numer. Math.56, 425–447

    Google Scholar 

  2. Berger, H. (1989): Finite-Element-Approximationen für transsonische Strömungen. PhD Thesis, Universität Stuttgart

  3. Berger, H., Warnecke, G., Wendland, W.L. (1990): Finite elements for transonic potential flows. Num. Meth. PDE6, 17–42

    Google Scholar 

  4. Bers, L. (1954): Existence and uniqueness of subsonic flow past a given profile. Comm. Pure. Appl. Math.7, 441–504

    Google Scholar 

  5. Bojarski, B. (1967): Subsonic flow of compressible fluid. Mathematical Problems in Fluid Mechanics. Polish Academy of Sciences, Warsaw

    Google Scholar 

  6. Bristeau, M.O. (1977): Application of optimal control theory to transonic flow computations by finite element methods. In Proceedings of the Third INRIA Symposium on Computing Methods in Applied Sciences and Engineering. pp. 103–124, Versailles, France

  7. Bristeau, M.O. (1980): Application of a finite element method to transonic flow problems using an optimal control approach. In: W., Kollmann, ed., Computational Fluid Dynamics. pp. 281–328, McGraw-Hill, New York

    Google Scholar 

  8. Bristeau, M.O., Glowinski, R., Periaux, J., Perrier, P., Pironneau, O. (1979): On the numerical solution of nonlinear problems in fluid dynamics by least squares and finite element methods I. Least squares formulations and conjugate gradient solution of the continuous problem. Comput. Methods Appl. Mech. Engrg.17/18, 619–657

    Google Scholar 

  9. Bristeau, M.O., Glowinski, R., Periaux, J., Perrier, P., Pironneau, O., Poirier, G. (1980): Application of optimal control and finite element methods to the calculation of transonic flows and incompressible flows. In: B. Hunt, ed., Numerical Methods in Applied Fluid Dynamics. pp. 203–312, Academic Press, New York

    Google Scholar 

  10. Bristeau, M.O., Glowinski, R., Periaux, J., Perrier, P., Pironneau, O., Poirier, G. (1982): Transonic flow simulations by finite elements and least squares methods. In: R.H. Gallagher, G. Carey, J.T. Oden, O.C. Zienkiewicz, eds., Finite Elements in Fluids vol. 4. pp. 453–482, Wiley, Chichester

    Google Scholar 

  11. Cavendish, J.C. (1974): Automatic triangulation of arbitrary planar domains for the finite element method. Int. J. Num. Meth. Eng.8, 679–696

    Google Scholar 

  12. Ciarlet, P.G. (1978): The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam

    Google Scholar 

  13. Feistauer, M., Nečas, J. (1985): On the solvability of transonic potential flow problems. Z. Anal. Anw.4, 305–329

    Google Scholar 

  14. Feistauer, M., Nečas, J. (1988): Remarks on the solvability of transonic flow problems. Manuscripta Math.61, 417–428

    Google Scholar 

  15. Glowinski, R. (1984): Numerical Methods for Nonlinear Variational Problems. Springer, Berlin Heidelberg New York

    Google Scholar 

  16. Göhner, U. (1991): Adaptive Finite-Element-Methoden für transsonische Strömungen. PhD Thesis, Universität Stuttgart

  17. Göhner, U. (1992): Adaptive finite element methods for transonic flows. (Manuscript)

  18. Göhner, U., Warnecke, G. (1992): A second order difference error indicator for adaptive transonic flow computations. (Manuscript)

  19. Göhner, U., Warnecke, G., Wendland, W. (1992): Error indicators for adaptive transonic flow computations. (Manuscript)

  20. Keyfitz, B., Warnecke, G. (1991): The existence of viscous profiles and admissibility for transonic shocks. Commun. Part. Diff. Eqns.16, 1197–1221

    Google Scholar 

  21. Lo, S.H. (1985): A new mesh generation scheme for arbitrary planar domains. Int. J. Num. Meth. Eng.21, 1403–1426

    Google Scholar 

  22. Morawetz, C.S. (1985): On a weak solution for a transonic flow problem. Comm. Pure Appl. Math.38, 797–817

    Google Scholar 

  23. Nečas, J. (1967): Les Méthodes Directes en Théorie des Équations Elliptiques. Masson, Paris

    Google Scholar 

  24. Nečas, J. (1986): Introduction to the Theory of Nonlinear Elliptic Equations. Wiley, New York

    Google Scholar 

  25. Rannacher, R., Scott, R. (1982): Some optimal error estimates for piecewise linear finite element approximations. Math. Comput.38, 437–445

    Google Scholar 

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The research reported in this article was supported by the Deutsche Forschungsgemeinschaft and the Volkswagen-Stiftung

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Göhner, U., Warnecke, G. A shock indicator for adaptive transonic flow computations. Numer. Math. 66, 423–448 (1993). https://doi.org/10.1007/BF01385706

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  • DOI: https://doi.org/10.1007/BF01385706

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