ANM for stationary Navier-Stokes equations and with Petrov-Galerkin formulation. (English) Zbl 1013.76046
Summary: This paper deals with the asymptotic numerical method (ANM) for solving nonlinear problems, with particular emphasis on stationary Navier-Stokes equations and Petrov-Galerkin formulation. ANM is a combination of a perturbation technique and a finite element method, allowing to transform a nonlinear problem into a succession of linear ones that admit the same tangent matrix. Earlier the method has been applied with success in nonlinear elasticity and fluid mechanics. In this paper, we apply the same kind of technique for solving Navier-Stokes equations with the so-called Petrov-Galerkin weighting. The main difficulty comes from the fact that the nonlinearity is no more quadratic, and it is not evident, in this case, to be able to compute a large number of terms of perturbation series. Several exampes from fluid mechanics are presented to demonstrate the performance of the method.
MSC:
| 76M10 | Finite element methods applied to problems in fluid mechanics |
| 76M45 | Asymptotic methods, singular perturbations applied to problems in fluid mechanics |
| 76D05 | Navier-Stokes equations for incompressible viscous fluids |
Keywords:
asymptotic numerical method; stationary Navier-Stokes equation; finite element method; Petrov-Galerkin weighting; perturbation seriesSoftware:
ABAQUSReferences:
| [1] | Thompson, International Journal of Solids and Structures 4 pp 757– (1968) · Zbl 0159.55701 · doi:10.1016/0020-7683(68)90054-1 |
| [2] | Perturbation Procedures in Non-linear Finite Element Structural Analysis, vol. 461. Springer: Berlin, 1975. |
| [3] | Noor, Computers and Structures 13 pp 31– (1981) · Zbl 0455.73080 · doi:10.1016/0045-7949(81)90106-1 |
| [4] | Noor, Computer Methods in Applied Mechanics and Engineering 28 pp 217– (1981) · Zbl 0466.73090 · doi:10.1016/0045-7825(81)90106-7 |
| [5] | Noor, Computers and Structures 16 pp 67– (1983) · Zbl 0498.73094 · doi:10.1016/0045-7949(83)90148-7 |
| [6] | Kawahara, International Journal for Numerical Methods in Engineering 10 pp 437– (1976) · Zbl 0345.76021 · doi:10.1002/nme.1620100213 |
| [7] | Cochelin, Computers and Structures 53 pp 1181– (1994) · Zbl 0918.73337 · doi:10.1016/0045-7949(94)90165-1 |
| [8] | Damil, International Journal for Engineering Science 28 pp 943– (1990) · Zbl 0721.73018 · doi:10.1016/0020-7225(90)90043-I |
| [9] | Azrar, International Journal for Numerical Methods in Engineering 36 pp 1251– (1993) · Zbl 0772.73039 · doi:10.1002/nme.1620360802 |
| [10] | Cochelin, Revue Européenne des Éléments Finis 3 pp 281– (1994) · Zbl 0810.73045 · doi:10.1080/12506559.1994.10511124 |
| [11] | Tri, Revue Européenne des Éléments Finis 5 pp 415– (1996) · Zbl 0924.76070 · doi:10.1080/12506559.1996.11509512 |
| [12] | Noor, AIAA Journal 23 pp 938– (1985) · Zbl 0562.73074 · doi:10.2514/3.9010 |
| [13] | Perturbation Methods in Fluid Mechanics. Academic Press: New York, 1964 (Version annotée 1975, Parabolic Press). · Zbl 0136.45001 |
| [14] | Perturbation Methods. Wiley: New York, 1973. · Zbl 0265.35002 |
| [15] | Riks, Computer Methods in Applied Mechanics and Engineering 47 pp 219– (1984) · Zbl 0535.73062 · doi:10.1016/0045-7825(84)90078-1 |
| [16] | Méthodes asymptotiques-numériques pour les fluides visqueux incompressibles et la détection de la bifurcation de Hopf. Thesis, Université de Metz, France. 1996. |
| [17] | Méthodes de résolution pour les fluides incompressibles. Thesis, Université de Technologie de Compiegne, France, 1995. |
| [18] | Calcul du problème de la houle non-linéaire et instationnaire par une méthode asymtotique numérique. Thesis, Université HASSAN II, Faculté des sciences BEN M’SIK Casablanca, Morocco, 1995. |
| [19] | The Finite Element Method (4th edn), vol. 2, Section 12.13. McGraw-Hill: New York, 1991. |
| [20] | Braikat, Revue Européenne des Eléments Finis 6 pp 337– (1997) · Zbl 0911.73062 · doi:10.1080/12506559.1997.10511274 |
| [21] | Potier-Ferry, Comptes Rendus de l’Académie des Sciences Paris, Série II b 324 pp 171– (1997) |
| [22] | Finite Element Methods of Navier-Stokes Equations. Reidel: Dordrecht, 1986. |
| [23] | Van Dyke, Quarterly Journal of Mechanics and Applied Mathematics 27 pp 423– (1974) · Zbl 0295.65066 · doi:10.1093/qjmam/27.4.423 |
| [24] | Najah, Archives of Computational Methods in Engineering 5 pp 3– (1998) · doi:10.1007/BF02736748 |
| [25] | Padé approximants Encyclopedia of Mathematics and its Applications. Cambridge University Press: Cambridge, (2nd edn). 1996. |
| [26] | Cochelin, International Journal for Numerical Methods in Engineering 37 pp 1187– (1994) · Zbl 0805.73076 · doi:10.1002/nme.1620370706 |
| [27] | An asymptotic-numerical method to compute bifurcating branches. In New Advances in Computational Structurals Mechanics, (eds). 1992; 117-131. |
| [28] | Elhage-Hussein, Revue Européenne des Eléments Finis 7 pp 119– (1998) · Zbl 0907.73080 · doi:10.1080/12506559.1998.11690469 |
| [29] | Hughes, Journal of Computational Physics 30 pp 1– (1979) · Zbl 0412.76023 · doi:10.1016/0021-9991(79)90086-X |
| [30] | Hughes, Computer Methods in Applied Mechanics and Engineering 32 pp 199– (1982) · Zbl 0497.76041 · doi:10.1016/0045-7825(82)90071-8 |
| [31] | Zienkiewicz, Computer Methods in Applied Mechanics and Engineering 78 pp 105– (1990) · Zbl 0708.76099 · doi:10.1016/0045-7825(90)90155-F |
| [32] | Ghia, Journal of Computational Physics 48 pp 387– (1982) · Zbl 0511.76031 · doi:10.1016/0021-9991(82)90058-4 |
| [33] | Multidimentional upwind schemes for the solution of the incompressibleNavier-Stokes equations. In Proceedings of the Third ECCOMAS Computational Fluid Dynamics Conference, (eds), Paris. Wiley: New York, 1996; 57-63. |
| [34] | A preconditioning technique for low-speed internal flows. In Proceedings of the 3rd ECCOMAS Computationnal Fluid Dynamics Conference, (eds), Paris. Wiley: New York, 1996; 924-930. |
| [35] | MARC. Theory and User Information, Vol. A, Version K7. Marc Analysis Research Corporation: Palo Alto, 1997; 11.1-11.30. |
| [36] | ABAQUS. Abaqus/Theory Manual. 2?Procedures/Nonlinear Solution Methods. Hibbitt, Karlsson & Sorensen: Pawtucket, 1999; 2.2.1-1. |
| [37] | Modélisation thermomécanique tridimensionnelle du laminage. Thesis, Ecole Nationale Supérieure des Mines de Paris, France, 1996. |
| [38] | Méthode asymptotique-numérique pour le calcul de bifurcations de Hopf et de solutions périodiques. Thesis, Université de Metz, France, 1995. |
| [39] | Price, Journal of Fluids Engineering 108 pp 193– (1986) · doi:10.1115/1.3242562 |
| [40] | Une méthode asymptotique numérique pour la dynamique non linéaire transitoire. In Proceedings of the 4th Colloque National en Calcul des Structures, Giens. (eds), Teckna, 1999; 239-244. |
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