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Galilean invariance and stabilized methods for compressible flows. (English) Zbl 1207.76094
Summary: In a recent work [the author et al., Comput. Methods Appl. Mech. Eng. 196, No. 4–6, 966–978 (2007; Zbl 1120.76334)], it was observed that lack of Galilean invariance led to catastrophic instabilities when stabilized methods were used in Lagrangian shock hydrodynamics computations. By means of an arbitrary Lagrangian-Eulerian (ALE) formulation, Galilean invariant SUPG operators were consistently derived in [the author, Comput. Methods Appl. Mech. Eng. 196, No. 4–6, 1108–1132 (2007; Zbl 1120.76333)], and their Lagrangian and Eulerian limits were compared to the most commonly used stabilized formulations. In the particular case of Eulerian meshes, it was shown that most of the SUPG operators designed to date for compressible flow computations are not invariant. However, due to the significant overhead of algebraic manipulations, the use in (the author, loc. cit.) of the referential form of the ALE equations made the presentation of the main ideas quite involved. The present paper addresses this particular issue, since the invariance analysis is presented with the aid of the intuitive current configuration reference frame, more familiar to computational fluid dynamicists.

MSC:
76M10 Finite element methods applied to problems in fluid mechanics
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
76N99 Compressible fluids and gas dynamics
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[1] Brooks, Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations, Computer Methods in Applied Mechanics and Engineering 32 pp 199– (1982) · Zbl 0497.76041
[2] Hughes, Computer Methods in Applied Mechanics and Engineering 45 pp 217– (1984) · Zbl 0542.76093
[3] Hughes, Multiscale phenomena: Green’s functions, the Dirichlet-to-Neumann formulation, subgrid-scale models, bubbles and the origin of stabilized methods, Computer Methods in Applied Mechanics and Engineering 127 pp 387– (1995) · Zbl 0866.76044
[4] Hughes, The variational multiscale method-a paradigm for computational mechanics, Computer Methods in Applied Mechanics and Engineering 166 pp 3– (1998) · Zbl 1017.65525
[5] Hughes, Large eddy simulation and the variational multiscale method, Computing and Visualization in Science 3 (47) pp 147– (2000) · Zbl 0998.76040
[6] Hughes, The multiscale formulation of large eddy simulation: decay of homogenous isotropic turbulence, Physics of Fluids 13 pp 505– (2001) · Zbl 1184.76236
[7] Hughes, Large eddy simulation of turbulent channel flows by the variational multiscale method, Physics of Fluids 13 (6) pp 1784– (2001) · Zbl 1184.76237
[8] Hughes, Encyclopedia of Computational Mechanics (2004)
[9] Scovazzi, Computer Methods in Applied Mechanics and Engineering 196 (4-6) pp 1108– (2007) · Zbl 1120.76333
[10] Tezduyar, Computation of moving boundaries and interfaces and stabilization parameters, International Journal for Numerical Methods in Fluids 43 pp 555– (2003) · Zbl 1032.76605
[11] Tezduyar, Stabilized finite element formulations for incompressible flow computations, Advances in Applied Mechanics 28 pp 1– (1992) · Zbl 0747.76069
[12] Tezduyar, Incompressible flow computations with stabilized bilinear and linear equal-order-interpolation velocity-pressure element, Computer Methods in Applied Mechanics and Engineering 95 pp 221– (1992) · Zbl 0756.76048
[13] Tezduyar, 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, Computer Methods in Applied Mechanics and Engineering 94 pp 339– (1992) · Zbl 0745.76044
[14] Tezduyar, 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, Computer Methods in Applied Mechanics and Engineering 94 pp 353– (1992) · Zbl 0745.76045
[15] Hughes, Proceedings of the XXI International Congress of Theoretical and Applied Mechanics (2004)
[16] Scovazzi, Computer Methods in Applied Mechanics and Engineering 196 (4-6) pp 966– (2007) · Zbl 1120.76333
[17] Belytschko, Nonlinear Finite Elements for Continua and Structures (2000)
[18] Donea, Computational Methods in Transient Analysis (1983)
[19] Donea, Encyclopedia of Computational Mechanics (2004)
[20] Scovazzi, Stabilized shock hydrodynamics: I. A Lagrangian method, Computer Methods in Applied Mechanics and Engineering 196 (4-6) pp 923– (2007) · Zbl 1120.76334
[21] Hauke, A unified approach to compressible and incompressible flows, Computer Methods in Applied Mechanics and Engineering 113 pp 389– (1994) · Zbl 0845.76040
[22] Hauke, A comparative study of different sets of variables for solving compressible and incompressible flows, Computer Methods in Applied Mechanics and Engineering 153 pp 1– (1998) · Zbl 0957.76028
[23] Hauke, Simple stabilizing matrices for the computation of compressible flows in primitive variables, Computer Methods in Applied Mechanics and Engineering 190 pp 6881– (2001) · Zbl 0996.76047
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