×

Implicit SUPG solution of Euler equations using edge-based data structures. (English) Zbl 1014.76039

Summary: We present an implicit, edge-based implementation of semi-discrete SUPG formulation with shock capturing for Euler equations in conservative variables. By disassembling the resulting finite element matrices into their edge contributions, sparse matrix coefficients, residuals and matrix-vector products needed in Krylov-update techniques are computed based on edge data structures. The resulting solution method requires less memory and CPU time than element-based implementations.

MSC:

76M10 Finite element methods applied to problems in fluid mechanics
76N15 Gas dynamics (general theory)
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Saad, Y., Iterative methods for sparse linear systems, (1996), PWS Publishing Boston · Zbl 1002.65042
[2] Johan, Z.; Hughes, T.J.R.; Shakib, F., A globally convergent matrix-free algorithm for implicit time-marching schemes arising in finite element analisys of fluids, Comput. meth. appl. mech. engrg., 87, 281-304, (1991) · Zbl 0760.76070
[3] Hughes, T.J.R.; Shakib, F.; Johan, Z., A multi-element group preconditioned GMRES algorithm for nonsymmetric systems arising in finite element analysis, Comput. meth. appl. mech. engrg., 65, 415-456, (1989) · Zbl 0687.76065
[4] Tezduyar, T.E.; Liou, J., Grouped element-by-element iteration schemes for incompressible flow simulations, Comp. phys. commun., 53, 441-453, (1989) · Zbl 0803.76065
[5] Liou, J.; Tezduyar, T., Clustered element-by-element computations for fluid flow, (), 167-187, (Chapter 9)
[6] Tezduyar, T.E.; Behr, M.; Aliabadi, S.K.; Mittal, S.; Ray, S., A new mixed preconditioning method for finite element computations, Comput. meth. appl. mech. engrg., 99, 27-42, (1992) · Zbl 0762.65060
[7] Kalro, V.; Tezduyar, T.E., Parallel iterative computational methods for 3D finite element flow simulations, Comp. assisted mech. engrg. sci., 5, 173-183, (1998) · Zbl 0952.76036
[8] Dutto, L.C.; Lepage, C.Y.; Habashi, W.G., Effect of the storage format of sparse linear systems on parallel CFD computations, Comput. meth. appl. mech. engrg., 188, 441-453, (2000) · Zbl 0971.76045
[9] Peraire, J.; Morgan, K.; Peiro, J., Unstructured grid methods for compressible flows, AGARD special course on unstructured grid methods for advection dominated flows, 787, 5.1, 5.39, (1992)
[10] Baum, J.D.; Luo, H.; Lohner, R., Edge-based finite element scheme for Euler equations, Aiaa j., 32, 1183-1190, (1994) · Zbl 0810.76037
[11] Peraire, J.; Lyra, P.R.M.; Morgan, K.; Peiro, J., TVD algorithms for the solution of the compressible Euler equations on unstructured meshes, Int. J. numer. meth. fluids, 19, 849-863, (1994) · Zbl 0824.76047
[12] Lohner, R., Edges, stars, superedges and chains, Comput. meth. appl. mech. engrg., 111, 255-263, (1994) · Zbl 0844.76078
[13] Martins, M.A.D.; Lyra, P.R.M.; Willmersdorf, R.; Coutinho, A.L.G.A., Parallel implementation of edge-based finite element schemes for compressible flows on unstructured meshes, (), 1-13
[14] Venkatakrishnan, V.; Mavriplis, D.J., Implicit solvers on unstructured meshes, J. computat. phys., 105, 83-91, (1993) · Zbl 0783.76065
[15] Luo, H.; Baum, J.D.; Lohner, R., A fast matrix-free implicit methods for compressible flows on unstructured grids, J. computat. phys., 146, 664-690, (1998) · Zbl 0931.76045
[16] Coutinho, A.L.G.A.; Martins, M.A.D.; Alves, J.L.D., Parallel iterative solution of finite element systems of equations employing edge-based data structures, (), 1-8
[17] Catabriga, L.; Martins, M.A.D.; Coutinho, A.L.G.A.; Alves, J.L.D., Clustered edge-by-edge preconditioners for non-symmetric finite element equations, (), 1-14
[18] Beau, G.J.L.; Tezduyar, T.E., Finite element computation of compressible flows with the SUPG formulation, (), 21, 27
[19] Almeida, R.C.; Galeão, A.C., An adaptative petrov – galerkin formulation for the compressible Euler and navier – stokes equations, Comput. meth. appl. mech. engrg., 129, 157-176, (1996) · Zbl 0865.76036
[20] Hirsh, C., ()
[21] Aliabadi, S.K.; Tezduyar, T.E., Parallel fluid dynamics computations in aerospace applications, Int. J. numer. meth. fluids, 21, 783-805, (1995) · Zbl 0862.76033
[22] F. Shakib, Finite element analysis of the compressible Euler and Navier-Stokes equations, Ph.D. thesis, Stanford University, Stanford, CA 94305, 1988
[23] Hughes, T.J.R.; Tezduyar, T.E., Finite element methods for first-order hyperbolic systems with particular emphasis on the compressible Euler equations, Comput. methods appl. mech. engrg., 45, 217-284, (1984) · Zbl 0542.76093
[24] Saad, Y.; Schultz, M.H., GMRES: a generalized minimal residual algorithm for solving nonsymetric linear systems, SIAM J. sci. statist. comput., 7, 856-869, (1986) · Zbl 0599.65018
[25] Catabriga, L.; Coutinho, A.L.G.A.; Almeida, R.C., Uma comparação de formulações estabilizadas de elementos finitos para a equação de Euler, (), 1-18
[26] Hauke, G.; Hughes, T.J.R., A comparative study of different sets of variables for solving compressible and incompressible flows, Comput. meth. appl. mech. engrg., 153, 1-44, (1998) · Zbl 0957.76028
[27] Beau, G.J.L.; Ray, S.E.; Aliabadi, S.K.; Tezduyar, T.E., SUPG finite element computation of compressible flows with the entropy and conservation variables formulations, Comput. meth. appl. mech. engrg., 104, 397-422, (1993) · Zbl 0772.76037
[28] Carey, G.F., Computational grids: generation, adaptation and solution strategies, (1997), Taylor and Francis Washington, DC, USA · Zbl 0955.74001
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.