×

Finite elements in fluids: stabilized formulations and moving boundaries and interfaces. (English) Zbl 1177.76202

Summary: We provide an overview of the finite element methods we developed for fluid dynamics problems. We focus on stabilized formulations and moving boundaries and interfaces. The stabilized formulations are the streamline-upwind/Petrov-Galerkin (SUPG) formulations for compressible and incompressible flows and the pressure-stabilizing/Petrov-Galerkin (PSPG) formulation for incompressible flows. These are supplemented with the discontinuity-capturing directional dissipation (DCDD) for incompressible flows and the shock-capturing terms for compressible flows. Determination of the stabilization and shock-capturing parameters used in these formulations is highlighted. Moving boundaries and interfaces include free surfaces, two-fluid interfaces, fluid-object and fluid-structure interactions, and moving mechanical components. The methods developed for this class of problems can be classified into two main categories: interface-tracking and interface-capturing techniques. The interface-tracking techniques are based on the deforming-spatial-domain/stabilized space-time (DSD/SST) formulation, where the mesh moves to track the interface. The interface-capturing techniques were developed for two-fluid flows. They are based on the stabilized formulation, over typically non-moving meshes, of both the flow equations and an advection equation. The advection equation governs the time-evolution of an interface function marking the interface location. We also describe some of the additional methods and ideas we introduced to increase the scope and accuracy of these two classes of techniques. Among them is the enhanced-discretization interface-capturing technique (EDICT), which was developed to increase the accuracy in capturing the interface. Also among them is the mixed interface-tracking/interface-capturing technique (MITICT), which was introduced for problems that involve both interfaces that can be accurately tracked with a moving-mesh method and interfaces that call for an interface-capturing technique.

MSC:

76M10 Finite element methods applied to problems in fluid mechanics
76-02 Research exposition (monographs, survey articles) pertaining to fluid mechanics
74F10 Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.)
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Hughes, T.J.R.; Brooks, A.N., A multi-dimensional upwind scheme with no crosswind diffusion, (), 19-35 · Zbl 0423.76067
[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 meth appl mech eng, 32, 199-259, (1982) · Zbl 0497.76041
[3] Tezduyar TE, Hughes TJR. Development of time-accurate finite element techniques for first-order hyperbolic systems with particular emphasis on the compressible Euler equations, NASA Technical Report NASA-CR-204772, NASA, 1982.
[4] Tezduyar TE, Hughes, TJR. Finite element formulations for convection dominated flows with particular emphasis on the compressible Euler equations. In: Proceedings of AIAA 21st Aerospace Sciences Meeting, AIAA Paper 83-0125, Reno, Nevada, 1983.
[5] Hughes, T.J.R.; Tezduyar, T.E., Finite element methods for first-order hyperbolic systems with particular emphasis on the compressible Euler equations, Comput meth appl mech eng, 45, 217-284, (1984) · Zbl 0542.76093
[6] Tezduyar, T.E., Stabilized finite element formulations for incompressible flow computations, Adv appl mech, 28, 1-44, (1992) · Zbl 0747.76069
[7] 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 meth appl mech eng, 95, 221-242, (1992) · Zbl 0756.76048
[8] Donea, J., A taylor – galerkin method for convective transport problems, Int J numer meth eng, 20, 101-120, (1984) · Zbl 0524.65071
[9] Johnson, C.; Navert, U.; Pitkäranta, J., Finite element methods for linear hyperbolic problems, Comput meth appl mech eng, 45, 285-312, (1984) · Zbl 0526.76087
[10] 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 multi-dimensional advective-diffusive systems, Comput meth appl mech eng, 63, 97-112, (1987) · Zbl 0635.76066
[11] Le Beau, G.J.; Tezduyar, T.E., Finite element computation of compressible flows with the SUPG formulation, (), 21-27
[12] Le Beau, G.J.; 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 eng, 104, 397-422, (1993) · Zbl 0772.76037
[13] Tezduyar, T.E.; Park, Y.J., Discontinuity capturing finite element formulations for nonlinear convection-diffusion-reaction equations, Comput meth appl mech eng, 59, 307-325, (1986) · Zbl 0593.76096
[14] Hughes, T.J.R.; Franca, L.P.; Balestra, M., A new finite element formulation for computational fluid dynamics: V. circumventing the babuška – brezzi condition: A stable petrov – galerkin formulation of the Stokes problem accommodating equal-order interpolations, Comput meth appl mech eng, 59, 85-99, (1986) · Zbl 0622.76077
[15] Franca, L.P.; Frey, S.L.; Hughes, T.J.R., Stabilized finite element methods: I. application to the advective-diffusive model, Comput meth appl mech eng, 95, 253-276, (1992) · Zbl 0759.76040
[16] Tezduyar, T.E.; Osawa, Y., Finite element stabilization parameters computed from element matrices and vectors, Comput meth appl mech eng, 190, 411-430, (2000) · Zbl 0973.76057
[17] Tezduyar TE. Calculation of the stabilization parameters in SUPG and PSPG formulations. In: Proceedings of the First South-American Congress on Computational Mechanics (CD-ROM), Santa Fe-Parana, Argentina, 2002.
[18] Tezduyar, T.E., Calculation of the stabilization parameters in finite element formulations of flow problems, (), 1-19 · Zbl 0747.76069
[19] Tezduyar TE. Adaptive determination of the finite element stabilization parameters. In: Proceedings of the ECCOMAS Computational Fluid Dynamics Conference 2001 (CD-ROM), Swansea, Wales, United Kingdom, 2001.
[20] Tezduyar, T.E., Computation of moving boundaries and interfaces and stabilization parameters, Int J numer meth fluids, 43, 555-575, (2003) · Zbl 1032.76605
[21] Catabriga L, Coutinho ALGA, Tezduyar TE. Finite element SUPG parameters computed from local matrices for compressible flows. In: Proceedings of the 9th Brazilian Congress of Engineering and Thermal Sciences, Caxambu, Brazil, 2002.
[22] Tezduyar, T.; Sathe, S., Stabilization parameters in SUPG and PSPG formulations, J comput appl mech, 4, 71-88, (2003) · Zbl 1026.76032
[23] Tezduyar, T.E., Finite element methods for fluid dynamics with moving boundaries and interfaces, (), Chapter 17 · Zbl 0848.76036
[24] Rispoli F, Borrelli P, Tezduyar TE. Discontinuity-capturing directional dissipation (DCDD) in computation of turbulent flows. In: Proceedings of the European Congress on Computational Methods in Applied Sciences and Engineering, ECCOMAS 2004 (CD-ROM), Jyvaskyla, Finland, 2004.
[25] Rispoli, F.; Borrelli, P.; Tezduyar, T.E., DCDD in finite element computation of turbulent flows, ()
[26] Tezduyar, T.E., Stabilized finite element methods for computation of flows with moving boundaries and interfaces, () · Zbl 0798.76037
[27] Tezduyar, T.E., Stabilized finite element methods for flows with moving boundaries and interfaces, HERMIS: the international journal of computer mathematics and its applications, 4, 63-88, (2003) · Zbl 1309.76135
[28] Tezduyar TE. Determination of the stabilization and shock-capturing parameters in supg formulation of compressible flows. In: Proceedings of the European Congress on Computational Methods in Applied Sciences and Engineering, ECCOMAS 2004 (CD-ROM), Jyvaskyla, Finland, 2004.
[29] Tezduyar TE, Senga M. Determination of the shock-capturing parameters in SUPG formulation of compressible flows. In: Proceedings of the 6th World Congress on Computational Mechanics (CD-ROM), Beijing, China, 2004. · Zbl 1122.76061
[30] 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 meth appl mech eng, 94, 339-351, (1992) · Zbl 0745.76044
[31] Tezduyar, T.E.; Behr, M.; Mittal, S.; 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 meth appl mech eng, 94, 353-371, (1992) · Zbl 0745.76045
[32] Tezduyar, T.E., Finite element methods for flow problems with moving boundaries and interfaces, Arch comput meth eng, 8, 83-130, (2001) · Zbl 1039.76037
[33] Tezduyar TE. Moving boundaries and interfaces. In: Franca LP, Tezduyar, TE, Masud A, editors, Finite Element Methods: 1970s and Beyond, 205-220, CIMNE, Barcelona, Spain, 2004.
[34] Hughes, T.J.R.; Hulbert, G.M., Space – time finite element methods for elastodynamics: formulations and error estimates, Comput meth appl mech eng, 66, 339-363, (1988) · Zbl 0616.73063
[35] Tezduyar TE, Sathe S, Keedy R, Stein K. Space – time techniques for finite element computation of flows with moving boundaries and interfaces. In: Gallegos S, Herrera I, Botello S, Zarate, F, Ayala G, editors. Proceedings of the III International Congress on Numerical Methods in Engineering and Applied Science, CD-ROM, 2004.
[36] Tezduyar TE, Sathe S, Keedy R, Stein K. Space-time finite element techniques for computation of fluid – structure interactions. Comput Meth Appl Mech Eng, in press. · Zbl 1118.74052
[37] Tezduyar, T.E.; Behr, M.; Mittal, S.; Johnson, A.A., Computation of unsteady incompressible flows with the finite element methods—space – time formulations, iterative strategies and massively parallel implementations, (), 7-24
[38] Tezduyar T. Finite element interface-tracking and interface-capturing techniques for flows with moving boundaries and interfaces. In: Proceedings of the ASME Symposium on Fluid-Physics and Heat Transfer for Macro- and Micro-Scale Gas-Liquid and Phase-Change Flows (CD-ROM), ASME Paper IMECE2001/HTD-24206, ASME, New York, 2001.
[39] Tezduyar, T.E., Stabilized finite element formulations and interface-tracking and interface-capturing techniques for incompressible flows, (), 221-239 · Zbl 1059.76038
[40] Stein, K.; Benney, R.; Kalro, V.; Tezduyar, T.E.; Leonard, J.; Accorsi, M., Parachute fluid – structure interactions: 3-D computation, Comput meth appl mech eng, 190, 373-386, (2000) · Zbl 0973.76055
[41] Stein, K.; Benney, R.; Tezduyar, T.; Potvin, J., Fluid – structure interactions of a cross parachute: numerical simulation, Comput meth appl mech eng, 191, 673-687, (2001) · Zbl 0999.76085
[42] Stein, K.R.; Benney, R.J.; Tezduyar, T.E.; Leonard, J.W.; Accorsi, M.L., Fluid – structure interactions of a round parachute: modeling and simulation techniques, J aircraft, 38, 800-808, (2001)
[43] Stein, K.; Tezduyar, T.; Kumar, V.; Sathe, S.; Benney, R.; Thornburg, E., Aerodynamic interactions between parachute canopies, J appl mech, 70, 50-57, (2003) · Zbl 1110.74690
[44] Tezduyar, T.; Aliabadi, S.; Behr, M., Enhanced-discretization interface-capturing technique, (), 1-6
[45] Tezduyar, T.; Aliabadi, S.; Behr, M., Enhanced-discretization interface-capturing technique (EDICT) for computation of unsteady flows with interfaces, Comput meth appl mech eng, 155, 235-248, (1998) · Zbl 0961.76046
[46] Farhat, C.; Lesoinne, M.; Maman, N., Mixed explicit/implicit time integration of coupled aeroelastic problems: three-field formulation, geometric conservation and distributed solution, Int J numer meth fluids, 21, 807-835, (1995) · Zbl 0865.76038
[47] Lesoinne, M.; Farhat, C., Geometric conservation laws for flow problems with moving boundaries and deformable meshes, and their impact on aeroelastic computations, Comput meth appl mech eng, 134, 71-90, (1996) · Zbl 0896.76044
[48] Lohner R, Yang C, Baum JD. Rigid and flexible store separation simulations using dynamic adaptive unstructured grid technologies. In: Proceedings of the First AFOSR Conference on Dynamic Motion CFD, New Brunswick, New Jersey, 1996.
[49] Garcia, J.; Onate, E.; Sierra, H.; Idelsohn, S., A stabilized numerical method for analysis of ship hydrodynamics, ()
[50] Onate, E.; Idelsohn, S.; Sacco, C.; Garcia, J., Stabilization of the numerical solution for the free surface wave equation in fluid dynamics, ()
[51] Onate E, Garcia J. A methodology for analysis of fluid – structure interaction accounting for free surface waves. In: Proceedings of European Conference on Computational Mechanics, Munich, Germany, 1999.
[52] Cruchaga, M.; Onate, E., A generalized streamline finite element approach for the analysis of incompressible flow problems including moving surfaces, Comput meth appl mech eng, 173, 241-255, (1999) · Zbl 0959.76041
[53] Tezduyar TE. Stabilization parameters and element length scales in SUPG and PSPG formulations. In: Book of Abstracts of An Euro Conference on Numerical Methods and Computational Mechanics, Miskolc, Hungary, 2002.
[54] Tezduyar T. Stabilization parameters and local length scales in SUPG and PSPG formulations. In: Proceedings of the Fifth World Congress on Computational Mechanics, On-line publication: http://wccm.tuwien.ac.at/, Paper-ID: 81508, Vienna, Austria, 2002.
[55] Tezduyar T. Interface-tracking and interface-capturing techniques for computation of moving boundaries and interfaces. In: Proceedings of the Fifth World Congress on Computational Mechanics, On-line publication: http://wccm.tuwien.ac.at/, Paper-ID: 81513, Vienna, Austria, 2002. · Zbl 1176.76076
[56] Hirt, C.W.; Nichols, B.D., Volume of fluid (VOF) method for the dynamics of free boundaries, J comput phys, 39, 201-225, (1981) · Zbl 0462.76020
[57] Tezduyar TE. Methods for computation of moving boundaries and interfaces. In: Proceedings of the European Congress on Computational Methods in Applied Sciences and Engineering, ECCOMAS 2004 (CD-ROM), Jyvaskyla, Finland, 2004.
[58] Tezduyar TE. Interface-tracking and interface-capturing techniques for finite element computation of moving boundaries and interfaces. Comput Meth Appl Mech Eng, in press. · Zbl 1176.76076
[59] Barbosa, H.J.C.; Hughes, T.J.R., The finite element method with Lagrange multipliers on the boundary: circumventing the babuska – brezzi condition, Comput meth appl mech eng, 85, 109-128, (1991) · Zbl 0764.73077
[60] Barbosa, H.J.C.; Hughes, T.J.R., Circumventing the babuska – brezzi condition in mixed finite element approximations of elliptic variational inequalities, Comput meth appl mech eng, 97, 193-210, (1992) · Zbl 0768.65033
[61] Tezduyar, T.E., Advanced computational techniques for moving boundaries and interfaces, () · Zbl 0747.76069
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.