Finite element modeling of multiscale transport phenomena

Finite Element Modeling of Multiscale Transport Phenomena

Volumn , Issue , 2010, Pages 1-250

  Nassehi, Vahid ^a   Parvazinia, Mahmoud ^b  

^a LOUGHBOROUGH UNIVERSITY   (United Kingdom)

^b IRAN POLYMER AND PETROCHEMICAL INSTITUTE   (Iran)

Author keywords

[No Author keywords available]

Indexed keywords

SYSTEMS ANALYSIS; TRANSPORT PROPERTIES;

ENVIRONMENTAL SYSTEMS ANALYSIS; MULTI-SCALE SYSTEM; MULTISCALE FINITE ELEMENT; MULTISCALE PROCESS; REACTION ENGINEERING; REACTION MECHANISM; STABILIZED FINITE ELEMENT; TRANSPORT PHENOMENA;

FINITE ELEMENT METHOD;

EID: 84967592309     PISSN: None     EISSN: None     Source Type: Book    
DOI: 10.1142/P657     Document Type: Book

References (94)

1. Dhatt, G. and Touzot, G., The Finite Element Displayed, John Wiley & Sons Inc., 1985.
2. Gerald, C.F. and Wheatley, P.O., Applied Numerical Analysis, 3rd ed., Addison-Wesley, Reading, Massachusetts, 1984.
3. Hinton, E. and Owen, D.J.R., Finite Element Programming, Academic Press, 1977.
4. Hood, P., Frontal solution program for unsymmetric matrices, Int. J. Numer. Meth. Eng., 1976; 10; 379-399.
5. Hughes, T.J.R. and Brooks, A.N., A multidimensional upwind scheme with no cross-wind diffusion, in Finite Element Methods for Convection Dominated Flows, AMD Vol. 34, TJR Hughes (Ed.), ASME, New York, 1979.
6. Irons, B. and Ahmad, S., Techniques of Finite Elements, Ellis Horwood/John Wiley & Sons, 1980.
7. Nassehi, V., Practical Aspects of Finite Element Modelling of Polymer Processing, John Wiley and Sons Ltd, Chichester, 2002.
8. Petera, J., Nassehi, V. and Pittman, J.F.T, Petrov-Galerkin methods on isoparametric bilinear and biquadratic elements tested for a scalar convection-diffusion problem, Int. J. Numer. Meth. Heat Fluid Flow, 1993; 3; 205-222.
9. Spiegel, M.R., Vector Analysis, Schaum’s Outline Series, McGraw-Hill Book Company, New York, 1974.
10. Zienkiewicz, O.C. and Morgan, K., Finite Elements and Approximation, John Wiley & Sons, New York, 1983.
11. Zienkiewicz, O.C. and Taylor, R.L., The Finite Element Method, 4th ed., Vol. 1 and 2, McGraw-Hill, London, 1994.
12. Brooks, A.N. and 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., 1982; 32; 199-259.
13. Huebner, K.H., Dewhirst, D.L., Smith, D.E. and Byrom, T.G., The Finite Element Method for Engineering, John Wiley & Sons Inc., 2001.
14. Hughes, T.J.R. and Brooks, A.N., A multidimensional upwind scheme with no crosswind diffusion, in Finite Element Methods for Convection Dominated Flows, AMD; T.J.R. Hughes (Ed.), ASME, New York, 1979; Vol. 34.
15. Li, B.Q., Discontinuous Finite Elements in Fluid Dynamics and Heat Transfer, Springer-Verlag, London, 2006.
16. Nassehi, V., Practical Aspects of Finite Element Modelling of Polymer Processing, John Wiley & Sons, Chichester, 2002.
17. Petera, J., Nassehi, V. and Pittman, J.F.T., Petrov-Galerkin methods on isoparametric bilinear and biquadratic elements tested for a scalar convection-diffusion problem, Int. J. Numer. Meth. Heat Fluid Flow, 1993; 3; 205-221.
18. Rao, S.S., The Finite Element Method in Engineering, Pergamon Press, 1989.
19. Tucker, C.L. III (Ed.). Computer Modeling for Polymer Processing, Hanser Publishers, Munich, 1989.
20. Alhumaizi, K., Henda, R. and Soliman, M., Numerical analysis of a reaction-diffusion-convection system, Comput. Chem. Eng., 2003; 27; 579-594.
21. Arnold, D.N., Brezzi, F. and Fortin, M., A stable finite element for the Stokes equations, Calcolo, 1984; 23; 337-344.
22. Baiocchi, C., Brezzi, F. and Franca, L.P., Virtual bubbles and the Galerkin/least squares method, Comput. Meth. Appl. Mech. Eng., 1993; 105; 125-142.
23. Brezzi, F., Bristeau, M.O., Franca, L.P., Mallet, M. and Roge, G., A relationship between stabilized finite element methods and the Galerkin method with bubble functions, Comput. Meth. Appl. Mech. Eng., 1992; 96; 117-129.
24. Brezzi, F. and Russo, A., Choosing bubbles for advection-diffusion problems, Math. Models Meth. Appl. Sci., 1994; 4; 571-587.
25. Brezzi, F., Franca L.P., Hughes, T.J.R. and Russo, A., B =g. Comput. Meth. Appl. Mech. Eng., 1997; 145; 329-339.
26. Brezzi, F., Franca, L.P. and Russo, A., Further considerations on residual- free bubbles for advective-diffusive equations, Comput. Meth. Appl. Mech. Eng., 1998; 166; 25-33.
27. Brezzi, F. and Marini, L.D., Augmented spaces, two-level methods, and stabilizing subgrids, Int. J. Numer. Meth. Fluids, 2002; 40; 31-46.
28. Brezzi, F., Hauke, G., Marini, L.D. and Sangalli, G., Link-cutting bubbles for convection-diffusion-reaction problems, Math. Models Meth. Appl. Sci., 2003; 13; 445-461.
29. Brezzi, F., Marini, L.D., and Russo, A., On the choice of a stabilizing subgrid for convection-diffusion problems, Comput. Meth. Appl. Mech. Eng., 2005; 194; 127-148.
30. Brooks, A. and Hughes, T.J.R., Streamline upwind/Petrov-Galerkin methods for advection dominated flows, Proc. Third Int. Conf. Finite Element Methods in Fluid Flow, Banff, Canada, 1980.
31. Brooks, A.N. and 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., 1982; 32; 199-259.
32. Burman, E. and Hansbo, P., Edge stabilization for Galerkin approximations of convection-diffusion-reaction problems, Comput. Meth. Appl. Mech. Eng., 2004; 193; 1437-1453.
33. Christie, I., Griffiths, D.F., Mitchell A.R. and Zienkiewicz, O.C., Finite element methods for second order differential equations with significant first derivatives, Int. J. Numer. Meth. Eng., 1976; 10; 1389-1396.
34. Codina, R., Comparison of some finite element methods for solving the diffusion-convection-reaction equation, Comput. Meth. Appl. Mech. Eng., 1998; 156; 185-210.
35. Donea, J. and Huerta, A., Finite Element Methods for Flow Problems, John Wiley & Sons, Chichester, 2003.
36. Farhat, C., Harari, I. and Franca, L.P., The discontinuous enrichment method, Comput. Meth. Appl. Mech. Eng., 2001; 190; 6455-6479.
37. Farhat, C., Harari, I. and Hetmaniuk U., A discontinuous Galerkin method with Lagrange multipliers for the solution of Helmholtz problems in the midfrequency regime. Comput. Meth. Appl. Mech. Eng., 2003a; 192; 1389-1419.
38. Farhat, C., Harari, I. and Hetmaniuk, U., The discontinuous enrichment method for multiscale analysis, Comput. Meth. Appl. Mech. Eng., 2003b; 192; 3195-3210.
39. Franca, L.P. and Russo, A., Deriving upwinding, mass lumping and selective reduced integration by residual free bubbles, Appl. Math. Lett., 1996; 9; 83-88.
40. Franca, L.P., Hughes, T.J.R. and Stenberg, R., Stabilized finite element methods for the stokes problem, in Incompressible Fluid Dynamics-Trends and Advances, Nicolaides, R.A. and Gunzberger, M.D. (Eds.), Cambridge University Press, Cambridge, 1993.
41. Franca, L.P. and Farhat, C., Bubble functions prompt unusual stabilized finite element methods, Comput. Meth. Appl. Mech. Eng., 1995; 123; 299-308.
42. Franca, L.P., Farhat, C., Macedo, A.P. and Lesoinne, M., Residual free bubbles for Helemholtz equation. Comput. Meth. Appl. Mech. Eng., 1997; 40; 4003-4009.
43. Franca, L.P. and Russo, A., Mass lumping emanating from residual free bubbles, Comput. Meth. Appl. Mech. Eng., 1997a; 142; 353-360.
44. Franca, L.P. and Russo, A., Unlocking with residual-free bubbles. Comput. Meth. Appl. Mech. Eng., 1997a; 142; 361-364.
45. Hauke, G., A simple subgrid scale stabilized method for the advection-diffusion-reaction equation, Comput. Meth. Appl. Mech. Eng., 2002; 191; 2925-2947.
46. Heinrich, J.C., Huyakorn, P.S., Zienkiewicz, O.C. and Mitchell, A.R., An upwind finite element scheme for two-dimensional convective transport equation, Int. J. Numer. Meth. Eng., 1977; 11; 134-143.
47. Hoffman, J., Johnson, C. and Bertoluzza, S., Subgrid modeling for convection-diffusion-reaction in one space dimension using a Haar Multiresolution analysis, Comput. Meth. Appl. Mech. Eng., 2005; 194; 19-44.
48. Hughes, T.J.R., A simple scheme for developing upwind finite elements, Int. J. Numer. Meth. Eng., 1978; 12; 1359-1365.
49. Hughes, T.J.R. and Brooks, A.N., A multidimensional upwind scheme with no crosswind diffusion, in Finite Element Methods for Convection Dominated Flows, T.J.R. Hughes (Ed.), AMD; vol. 34; ASME, New York, 1979.
50. Hughes, T.J.R., Multi scale phenomena, Green’s functions, the Dirichlet-to-Neumann formulation, subgrid scale models, bubbles and the origins of stabilized methods, Comput. Meth. Appl. Mech. Eng., 1995; 127; 381-401.
51. Hughes, T.J.R. and Stewart, J., A space time formulation for multiscale phenomena, Comput. Meth. Appl. Mech. Eng., 1996; 74; 217-229.
52. Hughes, T.J.R., Feijoo, G.R., Mazzei, L. and Quincy, J.B., The variational multiscale method - a paradigm for computational mechanics, Comput. Meth. Appl. Mech. Eng., 1998; 166; 3-24.
53. Juanes, R. and Patzek T.W., Multiscale-stabilized solutions to onedimensional systems of conservation laws, Comput. Meth. Appl. Mech. Eng., 2005; 194; 2781-2805.
54. Parvazinia, M., Nassehi, V., Wakeman, R.J. and Ghoreishy, M.H.R., Finite element modeling of flow through a porous medium between two parallel plates using the Brinkman equation, Transport Porous Media, 2006a; 63; 71-90.
55. Parvazinia, M., Nassehi, V. and Wakeman, R.J., Multi-scale finite element modeling of laminar steady flow through highly permeable porous media, Chem. Eng. Sci., 2006b; 61; 586-596.
56. Russo, A., Bubble stabilization of finite element methods for the linearized incompressible Navier-Stokes equations, Comput. Meth. Appl. Mech. Eng., 1996a; 132; 335-343.
57. Russo A., A posteriori error estimators via bubble functions, Math. Models Meth. Appl. Sci. 1996b; 6; 33-41.
58. Tezduyar, T.E. and Park, Y.J., Discontinuity-capturing finite element formulations for nonlinear convection-diffusion-reaction equations, Comput. Meth. Appl. Mech. Eng., 1986; 59; 307-325.
59. Bell, B.C. and Surana, K.S., A space-time coupled p-version least squares finite element formulation for unsteady two-dimensional Navier-Stokes equations, Int. J. Numer. Meth. Eng., 1996; 39; 2593-2618.
60. Brodkey, R.S. and Hershey, H.C., Transport Phenomena, McGraw Hill, 1988.
61. Donea, J. and Taylor, A., Galerkin method for convective transport problems, Int. J. Numer. Meth. Eng., 1984; 20; 101-120.
62. Donea, J., Roig, B. and Huerta, A., High order accurate time stepping schemes for convection-diffusion equation, Comput. Meth. Appl. Mech. Eng., 2000; 182; 249-275.
63. Donea, J. and Huerta, A., Finite Element Methods for Flow Problems, John Wiley & Sons, Chichester, 2003.
64. Franca, L.P., Hughes, T.J.R. and Stenberg, R., Stabilized finite element methods for the Stokes problem, in Nicolaides, R.A. and Gunzberger, M.D. (Eds.), Incompressible Fluid Dynamics - Trends and Advances, Cambridge University Press, Cambridge, 1993.
65. Franca, L.P., Farhat, C., Macedo, A.P. and Lesoinne, M., Residual free bubbles for Helemholtz equation, Comput. Meth. Appl. Mech. Eng., 1997; 40; 4003-4009.
66. Franca, L.P. and Russo, A., Deriving upwinding, mass lumping and selective reduced integration by residual free bubbles, Appl. Math. Lett., 1996; 9; 83-88.
67. Franca, L.P. and Russo, A., Unlocking with residual-free bubbles, Comput. Meth. Appl. Mech. Eng., 1997a; 142; 361-364.
68. Franca, L.P. and Russo, A., Mass lumping emanating from residual free bubbles, Comput. Meth. Appl. Mech. Eng., 1997b; 142; 353-360.
69. Franca, L.P. and Farhat, C., Bubble functions prompt unusual stabilized finite element methods, Comput. Meth. Appl. Mech. Eng., 1995; 123; 299-308.
70. Fried, I., Finite-element analysis of time-dependent phenomena, AIAA J, 1969; 7; 1170-1172.
71. Hughes, T.J.R. and Hulbert, G.M., Space-time finite element methods for elastodynamics: Fomulations and error estimates, Comput. Meth. Appl. Mech. Eng., 1998; 66; 339-363.
72. Hughes, T.J.R. and Stewart, J., A space time formulation for multiscale phenomena, J. Comput. Appl. Math., 1996; 74; 217-229.
73. Hughes, T.J.R., Multiscale phenomena, Green’s functions, the Dirichlet-to-Neumann formulation, subgrid scale models, bubbles and the origins of stabilized methods, Comput. Meth. Appl. Mech. Eng., 1995; 127; 381-401.
74. Lewis, R.W., Nithiarasu, P. and Seetharamu, K.N., Fundamentals of the Finite Element Method for Heat and Fluid Flow, John Wiley & Sons, Chichester, 2005.
75. Nassehi, V., Practical Aspects of Finite Element Modelling of Polymer Processing, John Wiley & Sons, Chichester, 2002.
76. Nassehi, V., Numerical modelling of tidal dynamics in a one-dimensional branching estuary, PhD thesis, University of Wales, Swansea, UK, 1981.
77. Nassehi, V. and Parvazinia, M., A multiscale finite element space-time discretization method for transient transport phenomena using bubble functions, Finite Elements in Analysis and Design, 2009; 45; 315-323.
78. Oden, J.T., A general theory of finite element, Part I and II, Int. J. Numer. Meth. Eng., 1969; 1; 205-221, 247-259.
79. Parvazinia, M., Nassehi, V. and Wakeman, R.J, Multiscale finite element modelling of laminar steady flow through highly permeable porous media, Chem. Eng. Sci., 2006; 61; 586-596.
80. Parvazinia, M. and Nassehi, V., Multiscale finite element modeling of diffusion-reaction equation using bubble functions with bilinear and triangular elements, Comput. Meth. Appl. Mech. Eng., 2007; 196; 1095-1107.
81. Parvazinia, M. and Nassehi, M., Using bubble functions in the multi-scale finite element modeling of the convection-diffusion-reaction equation, Int. Commun. Heat Mass Transfer, 2010; 37; 125-130.
82. Allan, F.M. and Hamdan, M.H., Fluid mechanics of the interface region between two porous layers, Appl. Math. Comput., 2002; 128; 37-43.
83. Bai, W., The quadrilateral ‘Mini’ finite element for the stokes problem, Comput. Meth. Appl. Mech. Eng., 1997; 143; 41-47.
84. Gerald, C.F. and Wheatley, P.O., Applied Numerical Analysis, Addison-Wesley, 1984.
85. Hsu, C.T. and Cheng, P., The Brinkman model for natural convection about a semi-infinite vertical flat plate in a porous medium, Int. J. Heat Mass Transfer, 1985; 28; 683-697.
86. Irons, B.M., A frontal solution program for finite element analysis, Int. J. Numer. Meth. Eng., 1970; 2; 5-32.
87. Kaviany, M., Non-Darcian effects on natural convection in porous media confined between horizontal cylinders, Int. J. Heat Mass Transfer, 1986; 29; 1513-1519.
88. Nassehi, V., Practical Aspects of Finite Element Modelling of Polymer Processing, John Wiley & Sons, Chichester, 2002.
89. Nield, D.A. and Bejan, A., Convection in Porous Media, Springer-Verlag, New York, 1992.
90. Parvazinia, M., Nassehi, V., Wakeman, R.J. and Ghoreishy, M.H.R., Finite element modelling of flow through a porous medium between two parallel plates using the Brinkman equation, Transport Porous Media, 2006a; 63; 71-90.
91. Parvazinia, M., Nassehi, V. and Wakeman, R.J., Multi-scale finite element modeling using bubble function method for a convection-diffusion problem, Chem. Eng. Sci., 2006b; 61; 2742-2751.
92. Pittman, J.F.T., Finite elements for field problems, in Computer Modelling for Polymer Processing, Tucker III, C.L. (Eds.), Hanser Publishers, Munich, 1989.
93. Reddy, J.N., Applied Functional Analysis and Variational Methods in Engineering, McGraw-Hill, New York, 1986.
94. Zienkiewicz, O.C. and Taylor, R.L., The Finite Element Method, McGraw-Hill, London, 1994.