Analysis of a FEM/BEM coupling method for transonic flow computations
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- by H. Berger, G. Warnecke and W. L. Wendland PDF
- Math. Comp. 66 (1997), 1407-1440 Request permission
Abstract:
A sensitive issue in numerical calculations for exterior flow problems, e.g. around airfoils, is the treatment of the far field boundary conditions on a computational domain which is bounded. In this paper we investigate this problem for two–dimensional transonic potential flows with subsonic far field flow around airfoil profiles. We take the artificial far field boundary in the subsonic flow region. In the far field we approximate the subsonic potential flow by the Prandtl–Glauert linearization. The latter leads via the Green representation theorem to a boundary integral equation on the far field boundary. This defines a nonlocal boundary condition for the interior ring domain. Our approach leads naturally to a coupled finite element/boundary element method for numerical calculations. It is compared with local boundary conditions. The error analysis for the method is given and we prove convergence provided the solution to the analytic transonic flow problem around the profile exists.References
- AGARD Advisory Report No. 211, Test Cases for Inviscid Flow Field Methods, Report of Fluid Dynamics Panel Working Group 07, 1985.
- Douglas N. Arnold and Wolfgang L. Wendland, The convergence of spline collocation for strongly elliptic equations on curves, Numer. Math. 47 (1985), no. 3, 317–341. MR 808553, DOI 10.1007/BF01389582
- Ivo Babuška and A. K. Aziz, Survey lectures on the mathematical foundations of the finite element method, The mathematical foundations of the finite element method with applications to partial differential equations (Proc. Sympos., Univ. Maryland, Baltimore, Md., 1972) Academic Press, New York, 1972, pp. 1–359. With the collaboration of G. Fix and R. B. Kellogg. MR 0421106
- E. B. Becker, G. F. Carey and J. T. Oden, Finite Elements, Vol. VI: Fluid Mechanics, Prentice–Hall Inc., Englewood Cliffs, New Jersey, 1984.
- H. Berger, "Finite–Element–Approximationen für transsonische Strömungen", Doctoral Thesis, Universität Stuttgart, Germany, 1989.
- Harald Berger, A convergent finite element formulation for transonic flow, Numer. Math. 56 (1989), no. 5, 425–447. MR 1021618, DOI 10.1007/BF01396647
- Harald Berger and Miloslav Feistauer, Analysis of the finite element variational crimes in the numerical approximation of transonic flow, Math. Comp. 61 (1993), no. 204, 493–521. MR 1192967, DOI 10.1090/S0025-5718-1993-1192967-6
- H. Berger, G. Warnecke and W. Wendland, "Finite Element–Berechnungen für transsonische Strömungen unter Berücksichtigung verschiedener Fernfeldrandbedingungen." In: Strömungen mit Ablösungen, DGLR–Bericht 88–05, Bonn (1988) 233-242.
- H. Berger, G. Warnecke, and W. Wendland, Finite elements for transonic potential flows, Numer. Methods Partial Differential Equations 6 (1990), no. 1, 17–42. MR 1034431, DOI 10.1002/num.1690060103
- H. Berger, G. Warnecke, and W. L. Wendland, Coupling of FEM and BEM for transonic flows, The mathematics of finite elements and applications (Uxbridge, 1993) Wiley, Chichester, 1994, pp. 323–350. MR 1291236
- Lipman Bers, Mathematical aspects of subsonic and transonic gas dynamics, Surveys in Applied Mathematics, Vol. 3, John Wiley & Sons, Inc., New York; Chapman & Hall, Ltd., London, 1958. MR 0096477
- Tadasi Nakayama, On Frobeniusean algebras. I, Ann. of Math. (2) 40 (1939), 611–633. MR 16, DOI 10.2307/1968946
- B. Bojarski, Subsonic flow of compressible fluid, Mathematical Problems in Fluid Mechanics, Państwowe Wydawnictwo Naukowe, Warsaw, 1967, pp. 9–31. MR 0229421
- Haïm Brézis and Guido Stampacchia, Une nouvelle méthode pour l’étude d’écoulements stationnaires, C. R. Acad. Sci. Paris Sér. A-B 276 (1973), A129–A132 (French). MR 315973
- M. O. Bristeau, O. Pironneau, R. Glowinski, J. Periaux, P. Perrier, and G. Poirier, Application of optimal control and finite element methods to the calculation of transonic flows and incompressible viscous flows, Numerical methods in applied fluid dynamics (Reading, 1978) Academic Press, London-New York, 1980, pp. 203–312. MR 657398
- M. O. Bristeau, R. Glowinski, J. Periaux, P. Perrier, O. Pironneau and G. Poirier, "Transonic flow simulations by finite elements and least squares methods," in Finite Elements in Fluids IV, R. H. Gallagher, G. Carey, J. T. Oden, and O. C. Zienkiewicz (eds.), Wiley, Chichester, 453-482 (1982).
- 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 flow simulations, Comput. Methods Appl. Mech. Engrg. 51 (1985), no. 1-3, 363–394. FENOMECH ’84, Part I, II (Stuttgart, 1984). MR 822749, DOI 10.1016/0045-7825(85)90039-8
- Philippe G. Ciarlet, The finite element method for elliptic problems, Studies in Mathematics and its Applications, Vol. 4, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1978. MR 0520174
- J.-F. Ciavaldini, M. Pogu, and G. Tournemine, Une nouvelle approche dans le plan physique pour le calcul d’écoulements subcritiques et stationnaires autour d’un profil portant, J. Mécanique 16 (1977), no. 2, 257–288 (French, with English summary). MR 452137
- J.-F. Ciavaldini, M. Pogu, and G. Tournemine, Existence and regularity of stream functions for subsonic flows past profiles with a sharp trailing edge, Arch. Rational Mech. Anal. 93 (1986), no. 1, 1–14. MR 822333, DOI 10.1007/BF00250842
- C. Coclici and W.L. Wendland, "On the treatment of the Kutta–Joukowski condition in transonic flow computations", in preparation (Preprint 95–14 MIA Univ. Stuttgart).
- Martin Costabel, Boundary integral operators on Lipschitz domains: elementary results, SIAM J. Math. Anal. 19 (1988), no. 3, 613–626. MR 937473, DOI 10.1137/0519043
- R. Courant and D. Hilbert, Methods of mathematical physics. Vol. II: Partial differential equations, Interscience Publishers (a division of John Wiley & Sons, Inc.), New York-London, 1962. (Vol. II by R. Courant.). MR 0140802
- Morgan Ward, Ring homomorphisms which are also lattice homomorphisms, Amer. J. Math. 61 (1939), 783–787. MR 10, DOI 10.2307/2371336
- M. Crouzeix and V. Thomée, The stability in $L_p$ and $W^1_p$ of the $L_2$-projection onto finite element function spaces, Math. Comp. 48 (1987), no. 178, 521–532. MR 878688, DOI 10.1090/S0025-5718-1987-0878688-2
- M. Dauge and M. Pogu, "Existence et régularité de la function potentiel pour des écoulements subcritiques s’établissant autour d’un corps à singularité conique," Ann. Fac. des Sci. de Toulouse IX, 213-242 (1988).
- M. Djaoua, A method of calculation of lifting flows around $2$-dimensional corner-shaped bodies, Math. Comp. 36 (1981), no. 154, 405–425. MR 606504, DOI 10.1090/S0025-5718-1981-0606504-7
- Herbert Federer, Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York, Inc., New York, 1969. MR 0257325
- Miloslav Feistauer, Jiří Felcman, Mirko Rokyta, and Zdeněk Vlášek, Finite-element solution of flow problems with trailing conditions, J. Comput. Appl. Math. 44 (1992), no. 2, 131–165. MR 1197680, DOI 10.1016/0377-0427(92)90008-L
- M. Feistauer, G. C. Hsiao, R. E. Kleinman, and R. Tezaur, Analysis and numerical realization of coupled BEM and FEM for nonlinear exterior problems, Inverse scattering and potential problems in mathematical physics (Oberwolfach, 1993) Methoden Verfahren Math. Phys., vol. 40, Peter Lang, Frankfurt am Main, 1995, pp. 47–73. MR 1319356
- M. Feistauer and J. Nečas, On the solvability of transonic potential flow problems, Z. Anal. Anwendungen 4 (1985), no. 4, 305–329 (English, with German and Russian summaries). MR 807140, DOI 10.4171/ZAA/155
- Miloslav Feistauer and Alexander Ženíšek, Finite element solution of nonlinear elliptic problems, Numer. Math. 50 (1987), no. 4, 451–475. MR 875168, DOI 10.1007/BF01396664
- Gilbert Strang and George J. Fix, An analysis of the finite element method, Prentice-Hall Series in Automatic Computation, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1973. MR 0443377
- F. I. Frankl and M. Keldysh, "Die äußere Neumannsche Aufgabe für nichtlineare elliptische Differentialgleichungen mit Anwendungen auf die Theorie der Flügel im kompressiblen Gas," Bull. Acad. Sci. URSS 12, 561-607 (1934).
- G. N. Gatica and G. C. Hsiao, "The coupling of boundary element and finite element methods for a nonlinear exterior boundary value problem," Numer. Math. 61, 171-214 (1992).
- R. Glowinski and O. Pironneau, "On the computation of transonic flows," in Funct. Anal. and Num. Anal., H. Fujita (ed.), Jap. Soc. Prom. Sci., Tokyo–Kyoto, 143-173 (1978).
- U. Göhner and G. Warnecke, A shock indicator for adaptive transonic flow computations, Numer. Math. 66 (1994), no. 4, 423–448. MR 1254397, DOI 10.1007/BF01385706
- George C. Hsiao and Wolfgang L. Wendland, A finite element method for some integral equations of the first kind, J. Math. Anal. Appl. 58 (1977), no. 3, 449–481. MR 461963, DOI 10.1016/0022-247X(77)90186-X
- Claes Johnson and J.-Claude Nédélec, On the coupling of boundary integral and finite element methods, Math. Comp. 35 (1980), no. 152, 1063–1079. MR 583487, DOI 10.1090/S0025-5718-1980-0583487-9
- Barbara Lee Keyfitz and Gerald G. Warnecke, The existence of viscous profiles and admissibility for transonic shocks, Comm. Partial Differential Equations 16 (1991), no. 6-7, 1197–1221. MR 1116859, DOI 10.1080/03605309108820795
- N. Kroll and R. K. Jain, Solution of Two–Dimensional Euler Equation Experience with a Finite Volume Code, DFVLR–Forschungsbericht 87–41, DFVLR Institut für Entwurfsaerodynamik, Braunschweig 1987.
- W. B. Berestetzki, E. M. Lifschitz, and L. P. Pitajewski, Lehrbuch der theoretischen Physik (“Landau-Lifschitz”). Band IV, 7th ed., Akademie-Verlag, Berlin, 1991 (German). Quantenelektrodynamik. [Quantum electrodynamics]; Translated from the Russian by Adolf Kühnel; Translation edited by Paul Ziesche; With a foreword by Ziesche and Kühnel. MR 1160351
- M. N. LeRoux, Résolution numérique du problème du potential dans le plan par une méthode variationelle d’éléments finis, Doctoral Thesis, University Rennes, 1974.
- Jan Mandel and Jindřich Nečas, Convergence of finite elements for transonic potential flows, SIAM J. Numer. Anal. 24 (1987), no. 5, 985–996. MR 909059, DOI 10.1137/0724064
- Saunders MacLane, Steinitz field towers for modular fields, Trans. Amer. Math. Soc. 46 (1939), 23–45. MR 17, DOI 10.1090/S0002-9947-1939-0000017-3
- Saunders MacLane and O. F. G. Schilling, Infinite number fields with Noether ideal theories, Amer. J. Math. 61 (1939), 771–782. MR 19, DOI 10.2307/2371335
- Cathleen S. Morawetz, Non-existence of transonic flow past a profile, Comm. Pure Appl. Math. 17 (1964), 357–367. MR 184522, DOI 10.1002/cpa.3160170308
- Cathleen S. Morawetz, On a weak solution for a transonic flow problem, Comm. Pure Appl. Math. 38 (1985), no. 6, 797–817. MR 812348, DOI 10.1002/cpa.3160380610
- François Murat, L’injection du cône positif de $H^{-1}$ dans $W^{-1,\,q}$ est compacte pour tout $q<2$, J. Math. Pures Appl. (9) 60 (1981), no. 3, 309–322 (French, with English summary). MR 633007
- J. Nečas, Compacite par Entropie et Ecoulements de Fluides, Lecture Notes Université de Charles et E.N-S., Masson, Paris (1989).
- Marc Pogu and Georges Tournemine, Sur une classe de problèmes monotones posés dans des ouverts non bornés, méthode d’approche, algorithmes de résolution et application, Rev. Roumaine Math. Pures Appl. 31 (1986), no. 4, 317–341 (French, with English summary). MR 854830
- Rolf Rannacher and Ridgway Scott, Some optimal error estimates for piecewise linear finite element approximations, Math. Comp. 38 (1982), no. 158, 437–445. MR 645661, DOI 10.1090/S0025-5718-1982-0645661-4
- Arthur Rizzi and Henri Viviand (eds.), Numerical methods for the computation of inviscid transonic flows with shock waves, Notes on Numerical Fluid Mechanics, vol. 3, Friedr. Vieweg & Sohn, Braunschweig, 1981. Papers from the GAMM Workshop. MR 680042
- J. Smoller, Shock Waves and Reaction–Diffusion Equations, Springer–Verlag, Berlin (1983).
- I.L. Sofronov and S.V. Tscyncov, "Application of the linear potential flow model to the artificial boundary conditions construction for the Euler equations." (Preprint No. 41 Nat. Center of Math. Simulation, Russian Acad. Sci. Moscow, 1991.) To appear.
- Gerald Warnecke, Admissibility of solutions to the Riemann problem for systems of mixed type—transonic small disturbance theory, Nonlinear evolution equations that change type, IMA Vol. Math. Appl., vol. 27, Springer, New York, 1990, pp. 258–284. MR 1074199, DOI 10.1007/978-1-4613-9049-7_{1}9
- J. Zierep, Theoretische Gasdynamik, Braun, Karlsruhe (1976).
Additional Information
- H. Berger
- Affiliation: Braunag F-1 TW4, Frankfurter Str 145, D-61476 Kronberg, Germany
- G. Warnecke
- Affiliation: Fakultät für Mathematik, Otto–von–Guericke–Universität Magdeburg, PF 4120, D–39016 Magdeburg, Germany
- Email: gerald.warnecke@mathematik.uni-magdeburg.de
- W. L. Wendland
- Affiliation: Mathematisches Institut A, Universität Stuttgart, Pfaffenwaldring 57, D-70569 Stuttgart, Germany
- Email: wendland@mathematik.uni-stuttgart.de
- Received by editor(s): August 13, 1993
- Received by editor(s) in revised form: September 18, 1995
- Additional Notes: The research reported in this paper was supported by the “Stiftung Volkswagenwerk”.
- © Copyright 1997 American Mathematical Society
- Journal: Math. Comp. 66 (1997), 1407-1440
- MSC (1991): Primary 65N30, 68N38, 76H05, 49M10, 35L67
- DOI: https://doi.org/10.1090/S0025-5718-97-00878-8
- MathSciNet review: 1432124
Dedicated: This work is dedicated to Professor Dr. Klaus Kirchgässner on the occasion of his 60th birthday