×

zbMATH — the first resource for mathematics

Weak imposition of Dirichlet boundary conditions in fluid mechanics. (English) Zbl 1115.76040
Summary: Weakly enforced Dirichlet boundary conditions are compared with strongly enforced conditions for boundary layer solutions of advection-diffusion equation and incompressible Navier-Stokes equations. It is found that weakly enforced conditions are effective and superior to strongly enforced conditions. The numerical tests involve low-order finite elements and a quadratic NURBS basis utilized in the isogeometric analysis approach. The convergence of mean velocity profile for a turbulent channel flow suggests that weak no-slip conditions behave very much like a wall function model, although the design of the boundary condition is based purely on numerical, rather than physical or empirical, conditions.

MSC:
76M10 Finite element methods applied to problems in fluid mechanics
76R99 Diffusion and convection
76D05 Navier-Stokes equations for incompressible viscous fluids
76F40 Turbulent boundary layers
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Abedi P, Patracovici B, Haber RB. A spacetime discontinuous Galerkin method for linearized elastodynamics with element-wise momentum balance. Comput Methods Appl Mech Eng, in press.
[2] Adjerid S, Massey TC. Superconvergence of discontinuous Galerkin solutions for a nonlinear scalar hyperbolic problem. Comput Methods Appl Mech Eng, in press. · Zbl 1124.65086
[3] Akin, J.E.; Tezduyar, T.E., Calculation of the advective limit of the SUPG stabilization parameter for linear and higher-order elements, Comput methods appl mech eng, 193, 1909-1922, (2004) · Zbl 1067.76557
[4] Antonietti PF, Buffa A, Perugia I. Discontinuous Galerkin approximation of the Laplace eigenproblem. Comput Methods Appl Mech Eng, in press. · Zbl 1168.65410
[5] Arnold, D.N.; Brezzi, F.; Cockburn, B.; Marini, L.D., Unified analysis of discontinuous Galerkin methods for elliptic problems, SIAM J numer anal, 39, 1749-1779, (2002) · Zbl 1008.65080
[6] Barth T. On discontinuous Galerkin approximations of Boltzmann moment systems with levermore closure. Comput Methods Appl Mech Eng, in press. · Zbl 1126.76027
[7] Baumann, C.E.; Oden, J.T., A discontinuous hp finite element method for convection – diffusion problems, Comput methods appl mech eng, 175, 311-341, (1999) · Zbl 0924.76051
[8] Bischoff, M.; Bletzinger, K.-U., Improving stability and accuracy of reissner – mindlin plate finite elements via algebraic subgrid scale stabilization, Comput methods appl mech eng, 193, 1491-1516, (2004)
[9] Bochev, P.B.; Gunzburger, M.D.; Shadid, J.N., On inf – sup stabilized finite element methods for transient problems, Comput methods appl mech eng, 193, 1471-1489, (2004) · Zbl 1079.76577
[10] Brezzi F, Cockburn B, Marini LD, Suli E. Stabilization mechanisms in discontinuous Galerkin finite element methods. Comput Methods Appl Mech Eng, in press. · Zbl 1125.65102
[11] Brezzi, F.; Hughes, T.J.R.; Süli, E., Variational approximation of flux in conforming finite element methods for elliptic partial differential equations: a model problem, Rend mat acc lincei, 9, 167-183, (2002)
[12] 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 methods appl mech eng, 32, 199-259, (1982) · Zbl 0497.76041
[13] Burman, E.; Hansbo, P., Edge stabilization for Galerkin approximations of convection – diffusion-reaction problems, Comput methods appl mech eng, 193, 1437-1453, (2004) · Zbl 1085.76033
[14] Calo VM. Residual-based multiscale turbulence modeling: finite volume simulation of bypass transition. PhD thesis, Department of Civil and Environmental Engineering, Stanford University, 2004.
[15] Chinosi C, Lovadina C, Marini LD. Nonconforming locking-free finite elements for Reissner-Mindlin plates. Comput Methods Appl Mech Eng, in press. · Zbl 1119.74047
[16] Chung, J.; Hulbert, G.M., A time integration algorithm for structural dynamics with improved numerical dissipation: the generalized-α method, J appl mech, 60, 371-375, (1993) · Zbl 0775.73337
[17] Ciarlet, P.G., The finite element method for elliptic problems, (1978), North-Holland Amsterdam · Zbl 0445.73043
[18] Cockburn B, Schotzau D, Wang J. Discontinuous Galerkin methods for incompressible elastic materials. Comput Methods Appl Mech Eng, in press. · Zbl 1128.74041
[19] Codina, R.; Soto, O., Approximation of the incompressible navier – stokes equations using orthogonal subscale stabilization and pressure segregation on anisotropic finite element meshes, Comput methods appl mech eng, 193, 1403-1419, (2004) · Zbl 1079.76579
[20] Coutinho, A.L.G.A.; Diaz, C.M.; Alvez, J.L.D.; Landau, L.; Loula, A.F.D.; Malta, S.M.C., Stabilized methods and post-processing techniques for miscible displacements, Comput methods appl mech eng, 193, 1421-1436, (2004) · Zbl 1079.76580
[21] Gravemeier, V.; Wall, W.A.; Ramm, E., A three-level finite element method for the instationary incompressible navier – stokes equations, Comput methods appl mech eng, 193, 1323-1366, (2004) · Zbl 1085.76038
[22] Grooss J, Hesthaven JS. A level-set discontinuous Galerkin method for free-surface flows. Comput Methods Appl Mech Eng, in press. · Zbl 1121.76035
[23] Harari, I., Stability of semidiscrete formulations for parabolic problems at small time steps, Comput methods appl mech eng, 193, 1491-1516, (2004) · Zbl 1079.76597
[24] Hauke, G.; Valiño, L., Computing reactive flows with a field Monte Carlo formulation and multi-scale methods, Comput methods appl mech eng, 193, 1455-1470, (2004) · Zbl 1079.76616
[25] Holmen, J.; Hughes, T.J.R.; Oberai, A.A.; Wells, G.N., Sensitivity of the scale partition for variational multiscale LES of channel flow, Phys fluids, 16, 3, 824-827, (2004) · Zbl 1186.76234
[26] Houston P, Schotzau D, Wihler TP. An hp-adaptive mixed discontinuous Galerkin FEM for nearly incompressible linear elasticity. Comput Methods Appl Mech Eng, in press. · Zbl 1118.74049
[27] Hughes, T.J.R.; Mazzei, L.; Jansen, K.E., Large-eddy simulation and the variational multiscale method, Comput visual sci, 3, 47-59, (2000) · Zbl 0998.76040
[28] Hughes, T.J.R.; Mazzei, L.; Oberai, A.A.; Wray, A.A., The multiscale formulation of large eddy simulation: decay of homogenous isotropic turbulence, Phys fluids, 13, 2, 505-512, (2001) · Zbl 1184.76236
[29] Hughes, T.J.R.; Oberai, A.A.; Mazzei, L., Large-eddy simulation of turbulent channel flows by the variational multiscale method, Phys fluids, 13, 6, 1784-1799, (2001) · Zbl 1184.76237
[30] Hughes, T.J.R.; Scovazzi, G.; Franca, L.P., Multiscale and stabilized methods, (), [Chapter 2]
[31] Hughes, T.J.R.; Calo, V.M.; Scovazzi, G., Variational and multiscale methods in turbulence, () · Zbl 1323.76032
[32] Hughes, T.J.R.; Cottrell, J.A.; Bazilevs, Y., Isogeometric analysis: CAD, finite elements, NURBS, exact geometry, and mesh refinement, Comput methods appl mech eng, 194, 4135-4195, (2005) · Zbl 1151.74419
[33] Hughes, T.J.R.; Engel, G.; Mazzei, L.; Larson, M., The continuous Galerkin method is locally conservative, J computat phys, 163, 2, 467-488, (2000) · Zbl 0969.65104
[34] Hughes TJR, Masud A, Wan J. A stabilized mixed discontinuous Galerkin method for Darcy flow. Comput Methods Appl Mech Eng, in press. · Zbl 1120.76040
[35] Hughes TJR, Wells GN, Wray AA. Energy transfers and spectral eddy viscosity of homogeneous isotropic turbulence: comparison of dynamic Smagorinsky and multiscale models over a range of discretizations. Technical report, ICES, The University of Texas at Austin, 2004. · Zbl 1187.76226
[36] Jansen, K.E.; Whiting, C.H.; Hulbert, G.M., A generalized-α method for integrating the filtered navier – stokes equations with a stabilized finite element method, Comput methods appl mech eng, 190, 305-319, (1999) · Zbl 0973.76048
[37] Kim, J.; Moin, P.; Moser, R., Turbulence statistics in fully developed channel flow at low Reynolds number, J fluid mech, 177, 133, (1987) · Zbl 0616.76071
[38] Koobus, B.; Farhat, C., A variational multiscale method for the large eddy simulation of compressible turbulent flows on unstructured meshes—application to vortex shedding, Comput methods appl mech eng, 193, 1367-1383, (2004) · Zbl 1079.76567
[39] Layton, W., Weak imposition of “no-slip” boundary conditions in finite element methods, Comput math appl, 38, 129-142, (1999) · Zbl 0953.76050
[40] Lin G, Karniadakis G. A discontinuous Galerkin method for two-temperature plazmas. Comput Methods Appl Mech Eng, in press.
[41] Masud, A.; Khurram, R.A., A multiscale/stabilized finite element method for the advection – diffusion equation, Comput methods appl mech eng, 193, 1997-2018, (2004) · Zbl 1067.76570
[42] Riviere B, Girault V. Discontinuous finite element methods for incompressible flows on subdomains with non-matching interfaces. Comput Methods Appl Mech Eng, in press. · Zbl 1121.76038
[43] Riviere, B.; Wheeler, M.F.; Girault, V., A priori error estimates for finite element methods based on discontinuous approximation spaces for elliptic problems, SIAM J numer anal, 39, 3, 902-931, (2001) · Zbl 1010.65045
[44] Romkes A, Prudhomme S, Oden JT. Convergence analysis of a discontinuous finite element formulation based on second order derivatives. Comput Methods Appl Mech Eng, in press. · Zbl 1125.65097
[45] Sun S, Wheeler MF. Anisotropic and dynamic mesh adaptation for discontinuous Galerkin methods applied to reactive transport. Comput Methods Appl Mech Eng, in press. · Zbl 1175.76096
[46] Taylor, C.A.; Hughes, T.J.R.; Zarins, C.K., Finite element modeling of blood flow in arteries, Comput methods appl mech eng, 158, 155-196, (1998) · Zbl 0953.76058
[47] Tejada-Martinez, A.E.; Jansen, K.E., On the interaction between dynamic model dissipation and numerical dissipation due to streamline upwind/petrov – galerkin stabilization, Comput methods appl mech eng, 194, 1225-1248, (2005) · Zbl 1091.76027
[48] Tezduyar, T.E.; Sathe, S., Enhanced-discretization space-time technique (EDSTT), Comput methods appl mech eng, 193, 1385-1401, (2004) · Zbl 1079.76585
[49] Tezduyar, T.E., Computation of moving boundaries and interfaces and stabilization parameters, Int J numer methods fluids, 43, 555-575, (2003) · Zbl 1032.76605
[50] Warburton T, Embree M. The role of the penalty in the local discontinuous Galerkin method for Maxwell’s eigenvalue problem. Comput Methods Appl Mech Eng, in press. · Zbl 1131.78011
[51] Wheeler, M.F., An elliptic collocation-finite element method with interior penalties, SIAM J numer anal, 15, 152-161, (1978) · Zbl 0384.65058
[52] Xu W. Private communication.
[53] Xu Y, Shu C-W. Local discontinuous Galerkin methods for the Kuramoto-Sivashinsky equations and the Ito-type coupled KdV equations. Comput Methods Appl Mech Eng, in press. · Zbl 1124.76035
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.