3D transient fixed point mesh adaptation for time-dependent problems: Application to CFD simulations. (English) Zbl 1158.76388
Summary: This paper deals with the adaptation of unstructured meshes in three dimensions for transient problems with an emphasis on CFD simulations. The classical mesh adaptation scheme appears inappropriate when dealing with such problems. Hence, another approach based on a new mesh adaptation algorithm and a metric intersection in time procedure, suitable for capturing and track such phenomena, is proposed. More precisely, the classical approach is generalized by inserting a new specific loop in the main adaptation loop in order to solve a transient fixed point problem for the mesh — solution couple. To perform the anisotropic metric intersection operation, we apply the simultaneous reduction of the corresponding quadratic form. Regarding the adaptation scheme, an anisotropic geometric error estimate based on a bound of the interpolation error is proposed. The resulting computational metric is then defined using the Hessian of the solution. The mesh adaptation stage (surface and volume) is based on the generation, by global remeshing, of a unit mesh with respect to the prescribed metric. A 2D model problem is used to illustrate the difficulties encountered. Then, 2D and 3D complexes and representative examples are presented to demonstrate the efficiency of this method.
MSC:
| 76M25 | Other numerical methods (fluid mechanics) (MSC2010) |
| 65M50 | Mesh generation, refinement, and adaptive methods for the numerical solution of initial value and initial-boundary value problems involving PDEs |
Keywords:
mesh adaptation; time-dependent problems; anisotropic metrics; geometric error estimate; CFD; Euler equations; unit meshSoftware:
HLLEReferences:
| [1] | Biswas, R.; Strawn, R., A new procedure for dynamic adaption of three-dimensional unstructured grids, Appl. Numer. Math., 13, 437-452 (1994) · Zbl 0793.76060 |
| [2] | Kallinderis, Y.; Vijayan, P., Adaptive refinement-coarsening scheme for three-dimensional unstructured meshes, AIAA J., 31, 8, 1440-1447 (1993) |
| [3] | D. Mavriplis, Adaptive meshing techniques for viscous flow calculations on mixed-element unstructured meshes, AIAA paper 97-0857.; D. Mavriplis, Adaptive meshing techniques for viscous flow calculations on mixed-element unstructured meshes, AIAA paper 97-0857. · Zbl 0969.76078 |
| [4] | Peraire, J.; Peiro, J.; Morgan, K., Adaptive remeshing for three-dimensional compressible flow computations, J. Comput. Phys., 103, 269-285 (1992) · Zbl 0764.76037 |
| [5] | Frey, P.; Alauzet, F., Anisotropic mesh adaptation for CFD computations, Comput. Methods Appl. Mech. Eng., 194, 48-49, 5068-5082 (2005) · Zbl 1092.76054 |
| [6] | Gruau, C.; Coupez, T., 3D tetrahedral, unstructured and anisotropic mesh generation with adaptation to natural and multidomain metric, Comput. Methods Appl. Mech. Eng., 194, 48-49, 4951-4976 (2005) · Zbl 1102.65122 |
| [7] | Li, X.; Shephard, M.; Beall, M., 3D anisotropic mesh adaptation by mesh modification, Comput. Methods Appl. Mech. Eng., 194, 48-49, 4915-4950 (2005) · Zbl 1090.76060 |
| [8] | Tam, A.; Ait-Ali-Yahia, D.; Robichaud, M.; Moore, M.; Kozel, V.; Habashi, W., Anisotropic mesh adaptation for 3D flows on structured and unstructured grids, Comput. Methods Appl. Mech. Eng., 189, 1205-1230 (2000) · Zbl 1005.76061 |
| [9] | Rausch, R.; Batina, J.; Yang, H., Spatial adaptation procedures on tetrahedral meshes for unsteady aerodynamic flow calculations, AIAA J., 30, 1243-1251 (1992) |
| [10] | Löhner, R., Three-dimensional fluid-structure interaction using a finite element solver and adaptive remeshing, Comput. Syst. Eng., 1, 2-4, 257-272 (1990) |
| [11] | Löhner, R.; Baum, J., Adaptive \(h\)-refinement on 3D unstructured grids for transient problems, Int. J. Numer. Meth. Fluids, 14, 1407-1419 (1992) · Zbl 0753.76099 |
| [12] | Speares, W.; Berzins, M., A 3D unstructured mesh adaptation algorithm for time-dependent shock-dominated problems, Int. J. Numer. Meth. Fluids, 25, 81-104 (1997) · Zbl 0881.76074 |
| [13] | de Sampaio, P.; Lyra, P.; Morgan, K.; Weatherill, N., Petrov-Galerkin solutions of the incompressible Navier-Stokes equations in primitive variables with adaptive remeshing, Comput. Methods Appl. Mech. Eng., 106, 143-178 (1993) · Zbl 0783.76053 |
| [14] | Wu, J.; Zhu, J.; Szmelter, J.; Zienkiewicz, O., Error estimation and adaptivity in Navier-Stokes incompressible flows, Comput. Mech., 6, 259-270 (1990) · Zbl 0699.76035 |
| [15] | Pain, C.; Humpleby, A.; de Oliveira, C.; Goddard, A., Tetrahedral mesh optimisation and adaptivity for steady-state and transient finite element calculations, Comput. Methods Appl. Mech. Eng., 190, 3771-3796 (2001) · Zbl 1008.76041 |
| [16] | Remacle, J.-F.; Li, X.; Shephard, M.; Flaherty, J., Anisotropic adaptive simulation of transient flows using discontinuous Galerkin methods, Int. J. Numer. Meth. Eng., 62, 899-923 (2005) · Zbl 1078.76042 |
| [17] | Alauzet, F.; George, P.-L.; Mohammadi, B.; Frey, P.; Borouchaki, H., Transient fixed point based unstructured mesh adaptation, Int. J. Numer. Meth. Fluids, 43, 6-7, 729-745 (2003) · Zbl 1032.76609 |
| [18] | Frey, P.; George, P.-L., Mesh Generation. Application to Finite Elements (2000), Hermès Science: Hermès Science Paris, Oxford |
| [19] | Debiez, C.; Dervieux, A., Mixed-element-volume MUSCL methods with weak viscosity for steady and unsteady flow calculations, Comput. Fluids, 29, 89-118 (2000) · Zbl 0949.76052 |
| [20] | Batten, P.; Clarke, N.; Lambert, C.; Causon, D., On the choice of wavespeeds for the HLLC riemann solver, SIAM J. Sci. Comput., 18, 6, 1553-1570 (1997) · Zbl 0992.65088 |
| [21] | P.-H. Cournède, C. Debiez, A. Dervieux, A positive MUSCL scheme for triangulations, Research Report INRIA, RR-3465, 1998.; P.-H. Cournède, C. Debiez, A. Dervieux, A positive MUSCL scheme for triangulations, Research Report INRIA, RR-3465, 1998. |
| [22] | Spiteri, R.; Ruuth, S., A new class of optimal high-order strong-stability-preserving time discretization methods, SIAM J. Numer. Anal., 40, 2, 469-491 (2002) · Zbl 1020.65064 |
| [23] | Castro-Diaz, M.; Hecht, F.; Mohammadi, B.; Pironneau, O., Anisotropic unstructured mesh adaptation for flow simulations, Int. J. Numer. Meth. Fluids, 25, 475-491 (1997) · Zbl 0902.76057 |
| [24] | P. Frey, About surface remeshing, in: Proceedings of the 9th International Meshing Rountable, New Orleans, LO, USA, 2000, pp. 123-136.; P. Frey, About surface remeshing, in: Proceedings of the 9th International Meshing Rountable, New Orleans, LO, USA, 2000, pp. 123-136. |
| [25] | P.-L. George, Tet meshing: construction, optimization and adaptation, in: Proceedings of the 8th International Meshing Rountable, South Lake Tao, CA, USA, 1999.; P.-L. George, Tet meshing: construction, optimization and adaptation, in: Proceedings of the 8th International Meshing Rountable, South Lake Tao, CA, USA, 1999. |
| [26] | Borouchaki, H.; Hecht, F.; Frey, P., Mesh gradation control, Int. J. Numer. Meth. Eng., 43, 6, 1143-1165 (1998) · Zbl 0944.76071 |
| [27] | Harten, A.; Lax, P.; Leer, B. V., On upstream differencing and Godunov-type schemes for hyperbolic conservation laws, SIAM Rev., 25, 1, 35-61 (1983) · Zbl 0565.65051 |
| [28] | Shu, C.; Osher, S., Efficient implementation of essentially non-oscillatory shock-capturing schemes, J. Comput. Phys., 77, 439-471 (1988) · Zbl 0653.65072 |
| [29] | Sod, G., A survey of several finite difference methods for systems of nonlinear hyperbolic conservation laws, J. Comput. Phys., 27, 1-31 (1978) · Zbl 0387.76063 |
| [30] | Sharp, D., An overview of Rayleigh-Taylor instability, Physica D, 12, 3-18 (1984) · Zbl 0577.76047 |
| [31] | Li, X.; Jin, B.; Glimm, J., Numerical study for the three-dimensional Rayleigh-Taylor instability through the TVD/AC scheme and parallel computation, J. Comput. Phys., 126, 343-355 (1996) · Zbl 0858.76055 |
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
