Zienkiewicz O.C., Taylor R.L. Vol. 3. The finite - tiera.ru

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The region studied was approximately 55 kilometres long and the mean value of the eddy di€usivity was of k ˆ 40 m s ÿ1 . The limiting time step for convection (considering eight components of tides) was 3.9 s. This limit was severely reduced to 0.1 s if the di€usion term was active and solved explicitly. The convective limit was recovered assuming an implicit solution with 3 ˆ 0:5. The comparisons of di€usion error between computations with 0.1 s and 3.9 s had a maximum di€usion error of 3.2% for the 3.9 s calculation, showing enough accuracy for engineering purposes, taking into account that the time stepping was increased 40 times, reducing dramatically the cost of computation. This reduction is fundamental when, in practical applications, the behaviour of the transported quantity must be computed for long-term periods, as was this problem, where the evolution of the salinity needed to be calculated for more than 60 periods of equivalent M 2 tides and for very di€erent initial conditions. In Fig. 7.14 we show by way of an example the dispersion of a continuous hot water discharge in an area of the Severn Estuary. Here we note not only the convection movement but also the di€usion of the temperature contours. References 1. M.B. Abbott. Computational Hydraulics: Elements of the Theory of Free Surface Flows, Pitman, London, 1979. 2. G.J. Haltiner and R.T. Williams. Numerical Prediction and Dynamic Meteorology, Wiley, New York, 1980. 3. O.C. Zienkiewicz, R.W. Lewis and K.G. Stagg (eds). Numerical Methods in O€shore Engineering, Wiley, Chichester, 1978. 4. J. Peraire. A ®nite element method for convection dominated ¯ows. Ph.D. thesis, University of Wales, Swansea, 1986. 5. J. Peraire, O.C. Zienkiewicz and K. Morgan. Shallow water problems: a general explicit formulation. Int. J. Num. Meth. Eng., 22, 547±74, 1986. 6. O.C. Zienkiewicz and J.C. Heinrich. A uni®ed treatment of steady state shallow water and two dimensional Navier±Stokes equations. Finite element penalty function approach. Comp. Meth. Appl. Mech. Eng., 17/18, 673±89, 1979. 7. M. Kawahara. On ®nite-element methods in shallow-water long-wave ¯ow analysis, in Computational Methods in Nonlinear Mechanics (ed. J.T. Oden), pp. 261±87, North- Holland, Amsterdam, 1980. 8. I.M. Navon. A review of ®nite element methods for solving the shallow water equations, in Computer Modelling in Ocean Engineering, 273±78, Balkema, Rotterdam, 1988. 9. J.J. Connor and C.A. Brebbia. Finite-Element Techniques for Fluid Flow, Newnes-Butterworth, London and Boston, 1976. 10. J.J. O'Brien and H.E. Hulburt. A numerical model of coastal upwelling. J. Phys. Oceanogr., 2, 14±26, 1972. 11. M. Crepon, M.C. Richez and M. Chartier. E€ects of coastline geometry on upwellings. J. Phys. Oceanogr., 14, 365±82, 1984. 12. M.G.G. Foreman. An analysis of two-step time-discretisations in the solution of the linearized shallow-water equations. J. Comp. Phys., 51, 454±83, 1983. 13. W.R. Gray and D.R. Lynch. Finite-element simulation of shallow-water equations with moving boundaries, in Proc. 2nd Conf. on Finite-Elements in Water Resources (eds C.A. Brebbia et al.), pp. 2.23±2.42, 1978. References 239
240 Shallow-water problems 14. T.D. Malone and J.T. Kuo. Semi-implicit ®nite-element methods applied to the solution of the shallow-water equations. J. Geophys. Res., 86, 4029±40, 1981. 15. G.J. Fix. Finite-element models for ocean-circulation problems. SIAM J. Appl. Math., 29, 371±87, 1975. 16. C. Taylor and J. Davis. Tidal and long-wave propagation, a ®nite-element approach. Computers and Fluids, 3, 125±48, 1975. 17. M.J.P. Cullen. A simple ®nite element method for meteorological problems. J. Inst. Math. Appl., 11, 15±31, 1973. 18. H.H. Wang, P. Halpern, J. Douglas, Jr. and I. Dupont. Numerical solutions of the onedimensional primitive equations using Galerkin approximations with localised basis functions. Mon. Weekly Rev., 100, 738±46, 1972. 19. I.M. Navon. Finite-element simulation of the shallow-water equations model on a limited area domain. Appl. Math. Modelling, 3, 337±48, 1979. 20. M.J.P. Cullen. The ®nite element method, in Numerical Methods Used in Atmosphere Models, Vol. 2, chap. 5, pp. 330±8, WMO/GARP Publication Series 17, World Meteorological Organisation, Geneva, Switzerland, 1979. 21. M.J.P. Cullen and C.D. Hall. Forecasting and general circulation results from ®niteelement models. Q. J. Roy. Met. Soc., 102, 571±92, 1979. 22. D.E. Hinsman, R.T. Williams and E. Woodward. Recent advances in the Galerkin ®niteelement method as applied to the meteorological equations on variable resolution grids, in Finite-Element Flow-Analysis (ed. T. Kawai), University of Tokyo Press, Tokyo, 1982. 23. I.M. Navon. A Numerov±Galerkin technique applied to a ®nite-element shallow-water equations model with enforced conservation of integral invariants and selective lumping. J. Comp. Phys., 52, 313±39, 1983. 24. I.M. Navon and R. de Villiers. GUSTAF, a quasi-Newton nonlinear ADI FORTRAN IV program for solving the shallow-water equations with augmented Lagrangians. Computers and Geosci., 12, 151±73, 1986. 25. A.N. Staniforth. A review of the application of the ®nite-element method to meteorological ¯ows, in Finite-Element Flow-Analysis (ed. T. Kawai), pp. 835±42, University of Tokyo Press, Tokyo, 1982. 26. A.N. Staniforth. The application of the ®nite element methods to meteorological simulations ± a review. Int. J. Num. Meth. Fluids, 4, 1-22, 1984. 27. R.T. Williams and O.C. Zienkiewicz. Improved ®nite element forms for shallow-water wave equations. Int. J. Num. Meth. Fluids, 1, 91±7, 1981. 28. M.G.G. Foreman. A two-dimensional dispersion analysis of selected methods for solving the linearised shallow-water equations. J. Comp. Phys., 56, 287±323, 1984. 29. I.M. Navon. FEUDX: a two-stage, high-accuracy, ®nite-element FORTRAN program for solving shallow-water equations. Computers and Geosci., 13, 225±85, 1987. 30. P. Bettess, C.A. Fleming, J.C. Heinrich, O.C. Zienkiewicz and D.I. Austin. A numerical model of longshore patterns due to a surf zone barrier, in 16th Coastal Engineering Conf., Hamburg, West Germany, October, 1978. 31. M. Kawahara, N. Takeuchi and T. Yoshida. Two step explicit ®nite element method for tsunami wave propagation analysis. Int. J. Num. Meth. Eng., 12, 331±51, 1978. 32. J.H.W. Lee, J. Peraire and O.C. Zienkiewicz. The characteristic Galerkin method for advection dominated problems ± an assessment. Comp. Meth. Appl. Mech. Eng., 61, 359±69, 1987. 33. D.R. Lynch and W. Gray. A wave equation model for ®nite element tidal computations. Computers and Fluids, 7, 207±28, 1979. 34. O. Daubert, J. Hervouet and A. Jami. Description on some numerical tools for solving incompressible turbulent and free surface ¯ows. Int. J. Num. Meth. Eng., 27, 3±20, 1989.
Page 1 and 2:
The Finite Element Method Fifth edi
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The Finite Element Method Fifth edi
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Dedication This book is dedicated t
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3.7 The performance of two- and sin
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...................................
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Volume 2: Solid and structural mech
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xiv Preface to Volume 3 this algori
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2 Introduction and the equations of
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10 Introduction and the equations o
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12 Introduction and the equations o
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14 Convection dominated problems We
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16 Convection dominated problems wh
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18 Convection dominated problems ch
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20 Convection dominated problems i.
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22 Convection dominated problems 2
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24 Convection dominated problems 2.
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26 Convection dominated problems ar
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28 Convection dominated problems Th
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30 Convection dominated problems co
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32 Convection dominated problems 2.
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34 Convection dominated problems ha
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36 Convection dominated problems an
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38 Convection dominated problems ac
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40 Convection dominated problems An
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42 Convection dominated problems U
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44 Convection dominated problems (a
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46 Convection dominated problems sp
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48 Convection dominated problems ag
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50 Convection dominated problems To
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52 Convection dominated problems C
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56 Convection dominated problems an
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60 Convection dominated problems Ma
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62 Convection dominated problems 47
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3 A general algorithm for compressi
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66 A general algorithm for compress
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92 Incompressible laminar ¯ow Cons
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94 Incompressible laminar ¯ow and,
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96 Incompressible laminar ¯ow V/V
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98 Incompressible laminar ¯ow The
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100 Incompressible laminar ¯ow 1.5
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102 Incompressible laminar ¯ow p 1
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104 Incompressible laminar ¯ow �
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106 Incompressible laminar ¯ow x 2
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108 Incompressible laminar ¯ow Thi
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114 Incompressible laminar ¯ow (a)
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116 Incompressible laminar ¯ow (b)
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118 Incompressible laminar ¯ow wit
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120 Incompressible laminar ¯ow The
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122 Incompressible laminar ¯ow Fig
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124 Incompressible laminar ¯ow and
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126 Incompressible laminar ¯ow U U
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128 Incompressible laminar ¯ow C L
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130 Incompressible laminar ¯ow are
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132 Incompressible laminar ¯ow sha
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134 Incompressible laminar ¯ow Fig
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136 Incompressible laminar ¯ow 28.
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138 Incompressible laminar ¯ow 71.
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140 Incompressible laminar ¯ow 114
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142 Incompressible laminar ¯ow 153
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144 Free surfaces, buoyancy and tur
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170 Compressible high-speed gas ¯o
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Page 231 and 232: 7 Shallow-water problems 7.1 Introd
Page 233 and 234: 220 Shallow-water problems reduced
Page 235 and 236: 222 Shallow-water problems Approxim
Page 237 and 238: 224 Shallow-water problems These si
Page 239 and 240: 226 Shallow-water problems h = 2 η
Page 241 and 242: 228 Shallow-water problems Surface
Page 243 and 244: 230 Shallow-water problems FL: 578
Page 245 and 246: 232 Shallow-water problems B Fig. 7
Page 247 and 248: 234 Shallow-water problems Time = 0
Page 249 and 250: 236 Shallow-water problems Fig. 7.1
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Page 255 and 256: 8 Waves Peter Bettess 8.1 Introduct
Page 257 and 258: 244 Waves the familiar form where
Page 259 and 260: 246 Waves of 10 nodes or thereabout
Page 261 and 262: 248 Waves and the Astley wave envel
Page 263 and 264: 250 Waves agreement with the analyt
Page 265 and 266: 252 Waves Fig. 8.3 General wave dom
Page 267 and 268: 254 Waves Relative errors (%) 10 8
Page 269 and 270: 256 Waves (The indirect boundary in
Page 271 and 272: 258 Waves where m is the number of
Page 273 and 274: 260 Waves Fig. 17.6 of Volume 1. La
Page 275 and 276: 262 Waves to using such coordinate
Page 277 and 278: 264 Waves r/a 5 4 3 2 1 0 0 2 4 6 8
Page 279 and 280: 266 Waves 8.16 Three-dimensional ef
Page 281 and 282: 268 Waves wave elevations in shallo
Page 283 and 284: 270 Waves integrals. Preliminary re
Page 285 and 286: 272 Waves 39. R.J. Astley. FE mode
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Page 289 and 290: 276 Computer implementation of the
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290 Computer implementation of the
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292 Appendix A Again substituting t
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294 Appendix B As usual the domain
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296 Appendix B 5. While the piecewi
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Appendix C Edge-based ®nite elemen
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Appendix D Multigrid methods It is
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Appendix E Boundary layer±inviscid
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304 Appendix E In Fig. E.2, Cp is t
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Author index Page numbers in bold r
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Gerdes, K. 259, 264, 265, 272 Ghia,
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Nakos, D.E. 146, 165 Narayana, P.A.
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Wu, J. 67, 90; 102, 108, 112, 137;
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316 Subject index Blurring of the s
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318 Subject index Convected coordin
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320 Subject index Elements ± cont.
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322 Subject index Flows: buoyancy d
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324 Subject index Incompressible St
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326 Subject index Metals: hot, 120
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328 Subject index Prescribed tracti
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330 Subject index Shock interaction
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332 Subject index Theorem ± cont.
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334 Subject index Wave ± cont. ste
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