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An analysis of the time integration algorithms for the finite element solutions of incompressible Navier–Stokes equations based on a stabilised formulation. (English) Zbl 1091.76521
Summary: This work is concerned with the analysis of time integration procedures for the stabilised finite element formulation of unsteady incompressible fluid flows governed by the Navier-Stokes equations. The stabilisation technique is combined with several different implicit time integration procedures including both finite difference and finite element schemes. Particular attention is given to the generalised-\(\alpha\) method and the linear discontinuous in time finite element scheme. The time integration schemes are first applied to two model problems, represented by a first-order differential equation in time and the one-dimensional advection-diffusion equation, and subjected to a detailed mathematical analysis based on the Fourier series expansion. In order to establish the accuracy and efficiency of the time integration schemes for the Navier-Stokes equations, a detailed computational study is performed of two standard numerical examples: unsteady flow around a cylinder and flow across a backward facing step. It is concluded that the semi-discrete generalised-\(\alpha\) method provides a viable alternative to the more sophisticated and expensive space-time methods for simulations of unsteady flows of incompressible fluids governed by the Navier-Stokes equations.

MSC:
76M10 Finite element methods applied to problems in fluid mechanics
76D05 Navier-Stokes equations for incompressible viscous fluids
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[1] Brezzi, F.; Fortin, M., Mixed and hybrid finite element methods, (1991), Springer New York · Zbl 0788.73002
[2] Brooks, A.N.; Hughes, T.J.R., Streamline-upwind/petrov – galerkin formulations for convection dominated flows with particular emphasis on the incompressible navier – stokes equations, Comput. methods appl. mech. engrg., 32, 199-259, (1982) · Zbl 0497.76041
[3] Hughes, T.J.R.; Franca, L.P.; Hulbert, G.M., A new finite element formulation for computational fluid dynamics: VIII. the Galerkin/least-squares method for advective – diffusive equations, Comput. methods appl. mech. engrg., 73, 173-189, (1989) · Zbl 0697.76100
[4] Tezduyar, T.E.; Mittal, S.; Ray, S.E.; Shih, R., Incompressible flow computations with stabilized bilinear and linear equal-order-interpolation velocity – pressure elements, Comput. methods appl. mech. engrg., 95, 221-242, (1992) · Zbl 0756.76048
[5] Hughes, T.J.R.; Franca, L.P.; Balestra, M., A new finite element formulation for computational fluid dynamics: V. circumventing the babuska – brezzi condition: A stable petrov – galerkin formulation of the Stokes problem accommodating equal-order interpolations, Comput. methods appl. mech. engrg., 59, 85-99, (1986) · Zbl 0622.76077
[6] Johnson, C.; Nävert, U.; Pitkäranta, J., Finite element methods for linear hyperbolic problems, Comput. methods appl. mech. engrg., 45, 285-312, (1984) · Zbl 0526.76087
[7] Brezzi, F.; Franca, L.P.; Hughes, T.J.R.; Russo, A., b=∫g, Comput. methods appl. mech. engrg., 145, 329-339, (1997) · Zbl 0904.76041
[8] Onate, E., A stabilized finite element method for incompressible viscous flows using a finite increment calculus formulation, Comput. methods appl. mech. engrg., 182, 355-370, (2000) · Zbl 0977.76050
[9] Zienkiewicz, O.C.; Codina, R., A general algorithm for compressible and incompressible flow. part I: the split characteristic based scheme, Int. J. numer. methods fluids, 20, 869-885, (1995) · Zbl 0837.76043
[10] Codina, R., Comparison of some finite element methods for solving the diffusion-convection-reaction equation, Comput. methods appl. mech. engrg., 156, 185-210, (1998) · Zbl 0959.76040
[11] Zienkiewicz, O.C.; Morgan, K.; Satya Sai, B.V.K.; Codina, R.; Vázquez, M., A general algorithm for compressible and incompressible flow. part II: tests on the explicit form, Int. J. numer. methods fluids, 20, 886-913, (1995) · Zbl 0837.76044
[12] Franca, L.P., Advances in stabilised methods in computational mechanics, Comput. methods appl. mech. engrg., 166, 1-182, (1998)
[13] Zienkiewicz, O.C.; Taylor, R.L., The finite element method, (2000), Butterworth-Heinemann Oxford · Zbl 0991.74002
[14] Chung, J.; Hulbert, G.M., A time integration algorithm for structural dynamics with improved numerical dissipation: the generalized α-method, J. appl. mech., 60, 371-375, (1993) · Zbl 0775.73337
[15] Jansen, K.E.; Whiting, C.H.; Hulbert, G.M., A generalized α-method for integrating the filtered navier – stokes equations with a stabilized finite element method, Comput. methods appl. mech. engrg., 190, 305-319, (2000) · Zbl 0973.76048
[16] Johnson, C., Numerical solution of partial differential equations by the finite element method, (1987), Cambridge University Cambridge
[17] Eriksson, K.; Estep, D.; Hansbo, P.; Johnson, C., Computational differential equations, (1996), Cambridge University Cambridge
[18] Thomée, V., Galerkin finite element methods for parabolic problems, (1997), Springer Berlin, Heidelberg · Zbl 0884.65097
[19] W. Dettmer, D. Perić, Time integration algorithms for stabilised finite element solutions of incompressible Navier-Stokes equations. Internal Report CR/1029, Department of Civil Engineering, University of Wales Swansea, December 2001
[20] Shakib, F.; Hughes, T.J.R.; Johan, Z., A new finite element formulation for computational fluid dynamics: X. the compressible Euler and navier – stokes equations, Comput. methods appl. mech. engrg., 89, 141-219, (1991) · Zbl 0838.76040
[21] Hughes, T.J.R., The finite element method: linear static and dynamic finite element analysis, (1987), Prentice-Hall Englewood Cliffs, NJ
[22] Gustafsson, B.; Kreiss, H.-O.; Oliger, J., Time dependent problems and difference methods, (1995), J. Wiley New York
[23] Shakib, F.; Hughes, T.J.R., A new finite element formulation for computational fluid dynamics: IX. Fourier analysis of space – time Galerkin/least-squares algorithms, Comput. methods appl. mech. engrg., 87, 35-58, (1991) · Zbl 0760.76051
[24] Gresho, P.M.; Sani, R.L., Incompressible flows and the finite element method, (1998), J. Wiley Chichester · Zbl 0941.76002
[25] Jansen, K.E.; Collis, S.S.; Shakib, F., A better consistency for low-order stabilized finite element methods, Comput. methods appl. mech. engrg., 174, 153-170, (1999) · Zbl 0956.76044
[26] Droux, J.-J.; Hughes, T.J.R., A boundary integral modification of the Galerkin/least-squares formulation for the Stokes problem, Comput. methods appl. mech. engrg., 113, 173-182, (1994) · Zbl 0845.76038
[27] Tezduyar, T.E.; Behr, M.; Liou, J., A new strategy for finite element computations involving moving boundaries and interfaces–the deforming-spatial-domain/space – time procedure: I. the concept and the preliminary numerical tests, Comput. methods appl. mech. engrg., 94, 339-351, (1992) · Zbl 0745.76044
[28] Tezduyar, T.E.; Behr, M.; Liou, J., A new strategy for finite element computations involving moving boundaries and interfaces–the deforming-spatial-domain/space – time procedure: II. computation of free-surface flows, two-liquid flows and flows with drifting cylinders, Comput. methods appl. mech. engrg., 94, 353-371, (1992) · Zbl 0745.76045
[29] Hughes, T.J.R.; Franca, L.P.; Mallet, M., A new finite element formulation for computational fluid dynamics: VI. convergence analysis of the generalized SUPG formulation for linear time dependent multidimensional advective – diffusive systems, Comput. methods appl. mech. engrg., 63, 97-112, (1987) · Zbl 0635.76066
[30] Pironneau, O., Finite element methods for fluids, (1989), J. Wiley Chichester · Zbl 0665.73059
[31] Marion, M.; Temam, R., Navier – stokes equations: theory and approximation, (), 503-689
[32] Schwab, C., P- and hp-finite element methods. theory and applications to solid and fluid mechanics, (1998), Oxford University Oxford · Zbl 0910.73003
[33] Tezduyar, T.E.; Osawa, Y., Finite element stabilization parameters computed from element matrices and vectors, Comput. methods appl. mech. engrg., 190, 411-430, (2000) · Zbl 0973.76057
[34] Masud, A.; Hughes, T.J.R., A space – time Galerkin/least-squares finite element formulation of the navier – stokes equations for moving domain problems, Comput. methods appl. mech. engrg., 146, 91-126, (1997) · Zbl 0899.76259
[35] Perić, D.; Slijepčević, S., Computational modelling of viscoplastic fluids based on a stabilised finite element method, Engrg. comput., 18, 577-591, (2001) · Zbl 1020.76029
[36] Simo, J.C.; Armero, F., Unconditional stability and long-term behaviour of transient algorithms for the incompressible navier – stokes and Euler equations, Comput. methods appl. mech. engrg., 111, 111-154, (1994) · Zbl 0846.76075
[37] Yee, H.C.; Torczynski, J.R.; Morton, S.A.; Visbal, M.R.; Sweby, P.K., On spurious behaviour of CFD simulations, Int. J. numer. methods fluids, 30, 675-711, (1999) · Zbl 0962.76080
[38] Gresho, P.M.; Dartling, D.K.; Torczynski, J.R.; Cliffe, K.A.; Winters, K.H.; Garratt, T.J.; Spence, A.; Goodrich, J.W., Is the steady viscous incompressible two-dimensional flow over a backward-facing step at re=800 stable?, Int. J. numer. methods fluids, 17, 501-541, (1993) · Zbl 0784.76050
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