×

zbMATH — the first resource for mathematics

A discontinuous Galerkin ALE method for compressible viscous flows in moving domains. (English) Zbl 0956.76046
Summary: We present a matrix-free discontinuous Galerkin method for simulating compressible viscous flows in two- and three-dimensional moving domains. To this end, we solve the Navier-Stokes equations in an arbitrary Lagrangian-Eulerian (ALE) framework. Spatial discretization is based on standard structured and unstructured grids, but using an orthogonal hierarchical spectral basis. The method is third-order accurate in time, and converges exponentially fast in space for smooth solutions. A novelty of the method is the use of a force-directed algorithm from graph theory that requires no matrix inversion to efficiently update the grid while minimizing distortions. We present several simulations using the method, including validation with published results for a pitching airfoil, and new results for flow past a three-dimensional wing subject to large flapping insect-like motion.

MSC:
76M10 Finite element methods applied to problems in fluid mechanics
76N15 Gas dynamics (general theory)
Software:
METIS
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Farhat, C; Lesoinne, M; LeTallec, P, Load and motion transfer algorithms for fluid/structure interaction problems with non-matching discrete interfaces: momentum and energy conservation, optimal discretization and application to aeroelasticity, Comput. methods appl. mech. eng., 157, 95, (1998) · Zbl 0951.74015
[2] Tezduyar, T; Behr, M; Liu, 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. eng., 94, 339, (1992) · Zbl 0745.76044
[3] Sherwin, S.J; Karniadakis, G.E, Tetrahedral hp finite elements: algorithms and flow simulations, J. comput. phys., 122, 191, (1996) · Zbl 0847.76038
[4] Lomtev, I; Quillen, C; Karniadakis, G.E, Spectral/hp methods for viscous compressible flows on unstructured 2D meshes, J. comput. phys., 144, 325, (1998) · Zbl 0929.76095
[5] Dubiner, M, Spectral methods on triangles and other domains, J. sci. comput., 6, 345, (1991) · Zbl 0742.76059
[6] Sherwin, S.J, Hierarchical hp finite elements in hybrid domains, Finite element anal. design, 27, 109, (1997) · Zbl 0896.65074
[7] Cockburn, B; Shu, C.-W, The local discontinuous Galerkin for time dependent convection-diffusion systems, SIAM J. numer. anal., 35, 2440, (1998) · Zbl 0927.65118
[8] Cockburn, B; Shu, C.-W, TVB runge – kutta discontinuous method for conservation laws, V, J. comput. phys., 141, 199, (1998)
[9] Cockburn, B; Shu, C.-W, TVB runge – kutta local projection discontinuous Galerkin finite element method for conservation laws. II. general framework, Math. comp., 52, 411, (1989) · Zbl 0662.65083
[10] Cockburn, B; Lin, S.-Y; Shu, C.-W, TVB runge – kutta local projection discontinuous Galerkin finite element method for conservation laws. III. one-dimensional systems, J. comput. phys., 84, 90, (1989) · Zbl 0677.65093
[11] Cockburn, B; Hou, S; Shu, C.-W, The runge – kutta local projection discontinuous Galerkin finite element method for conservation laws. IV. the multi-dimensional case, J. comput. phys., 54, 545, (1990) · Zbl 0695.65066
[12] B. Cockburn, and, C.-W. Shu, P1-RKDG method for two-dimensional Euler equations of gas dynamics, in, Proc. Fourth Int. Symp. on CFD, UC Davis, 1991.
[13] Biswas, R; Devine, K; Flaherty, J, Parallel, adaptive finite element methods for conservation laws, Appl. numer. math., 14, 255, (1994) · Zbl 0826.65084
[14] Bassi, F; Rebay, S, A high-order accurate discontinuous finite element method for the numerical solution of the compressible navier – stokes equations, J. comput. phys., 131, 267, (1997) · Zbl 0871.76040
[15] F. Bassi, S. Rebay, M. Savini, G. Marioti, and, S. Pedinotti, A high-order accurate discontinuous finite element method for inviscid and viscous turbomachinery flows, in, Proceedings of the Second European Conference on Turbomachinery Fluid Dynamics and Thermodynamics, Antwerpen, Belgium, March 5-7, 1997.
[16] Bey, K.S; Patra, A; Oden, J.T, Hp version discontinuous Galerkin methods for hyperbolic conservation laws: A parallel adaptive strategy, Int. J. numer. methods eng., 38, 3889, (1995) · Zbl 0855.65106
[17] Bey, K.S; Patra, A; Oden, J.T, Hp version discontinuous Galerkin methods for hyperbolic conservation laws, Comput. methods appl. mech. eng., 133, 259, (1996) · Zbl 0894.76036
[18] Oden, J.T; Babuska, I; Baumann, C.E, A discontinuous hp finite element method for diffusion problems, J. comput. phys., 146, 491, (1998) · Zbl 0926.65109
[19] C. E. Baumann, and, J. T. Oden, A discontinuous hp finite element method for the solution of the Euler and Navier-Stokes equations, Int. J. Numer. Methods Fluids, in press. · Zbl 0985.76048
[20] Baumann, C.E; Oden, J.T, A discontinuous hp finite element method for convection-diffusion problems, Comput. methods appl. mech. eng., 175, 311, (1999) · Zbl 0924.76051
[21] Hughes, T.J.R; Liu, W.K; Zimmerman, T.K, Lagrangian – eulerian finite element formulation for incompressible viscous flows, Comput. methods appl. mech. eng., 29, 329, (1981) · Zbl 0482.76039
[22] Nomura, T; Hughes, T.J.R, An arbitrary lagrangian – eulerian finite element method for interaction of fluid and a rigid body, Comput. methods appl. mech. eng., 95, 115, (1992) · Zbl 0756.76047
[23] Hirt, C.W; Amsden, A.A; Cook, H.K, An arbitrary lagrangian – eulerian computing method for all flow speeds, J. comput. phys., 14, 27, (1974) · Zbl 0292.76018
[24] Donea, J; Giuliani, S; Halleux, J.P, An arbitrary lagrangian – eulerian finite element method for transient dynamic fluid-structure interactions, Comput. methods appl. mech. eng., 33, 689, (1982) · Zbl 0508.73063
[25] Ho, L.-W, A Legendre spectral element method for simulation of incompressible unsteady free-surface flows, (1989)
[26] C. S. Venkatasubban, A new finite element formulation for ALE Arbitrary Lagrangian Eulerian compressible fluid mechanics, Int. J. Eng. Sci. 33, 1743, 1995. · Zbl 0899.76263
[27] Lohner, R; Yang, C, Improved ale mesh velocities for moving bodies, Comm. numer. methods eng. phys., 12, 599, (1996) · Zbl 0858.76042
[28] Di Battista, G; Eades, P; Tamassia, R; Tollis, I.G, Graph drawing, (1998)
[29] Karniadakis, G.E; Sherwin, S.J, Spectral/hp element methods for CFD, (1999)
[30] Jiang, G; Shu, C.W, On a cell entropy inequality for discontinuous Galerkin methods, Math. comp., 62, 531, (1994) · Zbl 0801.65098
[31] Johnson, C; Pitkaranta, J, An analysis of the discontinuous Galerkin method for a scalar hyperbolic equation, Math. comp., 46, 1, (1986) · Zbl 0618.65105
[32] Jaffre, J; Johnson, C; Szepessy, A, Convergence of the discontinuous Galerkin finite element method for hyperbolic conservation laws, Math. models methods appl. sci., 5, 367, (1995) · Zbl 0834.65089
[33] B. Koobus, and, C. Farhat, Second-order schemes that satisfy GCL for flow computations on dynamic grids, in, AIAA 98-0113, 36th AIAA Aerospace Sciences Meeting and Exhibit, Reno, NV, January 12-15, 1998.
[34] H. Guillard, and, C. Farhat, On the significance of the GCL for flow computations on moving meshes, in, AIAA 99-0793, 37th AIAA Aerospace Sciences Meeting and Exhibit, Reno, NV, January 11-14, 1999.
[35] Giannakouros, I.G, Spectral element/flux-corrected methods for unsteady compressible viscous flows, (1994)
[36] Batina, J.T, Unsteady Euler airfoil solutions using unstructed dynamic meshes, Aiaa j, 28, 1381, (1990)
[37] Rausch, R.D; Batina, J.T; Yang, H.T.Y, Three-dimensional time-marching aeroelastic analyses using an unstructured-grid Euler method, Aiaa j., 31, 1626, (1994)
[38] Blom, F.J; Leyland, P, Analysis of fluid-structure interaction by means of dynamic unstructured meshes, Trans. ASME, 120, 792, (1998)
[39] Lomtev, I, Discontinuous spectral/hp element methods for high speed flows, (1999)
[40] Visbal, M.R; Shang, J.S, Investigation of the flow structure around a rapidly pitching airfoil, Aiaa j., 27, 1044, (1989)
[41] Beam, R; Warming, R, An implicit factored scheme for the compressible navier – stokes equations, Aiaa j., 16, 393, (1978) · Zbl 0374.76025
[42] Liu, H; Kawachi, K, A numerical study of insect flight, J. comput. phys., 146, 124, (1998) · Zbl 0929.76092
[43] Karypis, G; Kumar, V, METIS: unstructured graph partitioning and sparse matrix ordering system version 2.0, (1995)
[44] Gottlieb, D; Orszag, S.A, Numerical analysis of spectral methods, (1977)
[45] Baumann, C.E, An hp-adaptive discontinuous finite element method for computational fluid dynamics, (1997)
[46] Johnson, C, Numerical solution of partial differential equations by the finite element method, (1994)
[47] H. Tran, B. Koobus, and, C. Farhat, Numerical solution of vortex dominated flow problems with moving grids, in, AIAA 98-0766, 36th AIAA Aerospace Sciences Meeting and Exhibit, Reno, NV, January 12-15, 1998.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.