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

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νtang = νtroll = 28.73 cm/s k ∂T/∂n = α2 (T–322) y, ν All dimensions in cm k ∂T/∂n = α2 (T–295) 30.2 ∂T/∂n = 0 k ∂T/∂n = α 1 (T–295) T = T 1 0.889 C L x, u 0.633 ν = 0 ∂T/∂n = 0 2.25 3.665 2.25 (a) Geometry Non-newtonian ¯ows ± metal and polymer forming 125 T = 460 K Temperature contours for entry T = 400 K at intervals ∆T = 4.0 K T = 400 K T = 722 K Temperature contours for entry T = 700 K at intervals ∆T = 1.5 K T = 700 K Horizontal Vertical 9 x = 2.25 cm 1 9 x = 3.25 cm 1 9 x = 4.9575 cm 1 9 x = 5.915 cm 1 9 x = 6.15 cm 1 20 30 1.0 –1.0 –3.0 cm/s (b) Velocity profiles Nodal points (c) Temperature distribution for different entry temperatures Fig. 4.22 Steady-state rolling process with thermal coupling. 93
126 Incompressible laminar ¯ow U U (a) t = 0 60% (c) t = 30∆t u∆t Indeed, both the coordinate and thermal updating can make use iteratively of the solution on the updated mesh to increase accuracy. However, it must be noted that any continuous mesh updating will soon lead to unacceptable meshes and some form of remeshing is necessary. In the example of Fig. 4.23, 89 in which ideal plasticity was assumed together with isothermal behaviour, it is necessary only to keep track of boundary movements. As temperature and other state variables do not enter the problem the remeshing can be done simply ± in the case shown by keeping the same vertical lines for the mesh position. However, in the example of Fig. 4.24 showing a more realistic problem, 131;132 when a new mesh is created an interpolation of all the state parameters from the old to the new mesh positions is necessary. In such problems it is worthwhile to strive to obtain discretization errors within speci®ed bounds and to remesh adaptively when these errors are too large. We have discussed the problem of adaptive remeshing for linear problems in Chapter 15 of Volume 1. In the present examples similar methods have been adopted with success 133;134 and in Fig. 4.24 we show how remeshing proceeds during the U U 30% (b) t = 15∆t 90% (d) t = 45∆t Fig. 4.23 Punch indentation problem (penalty function approach). 89 Updated mesh and surface pro®le with 24 isoparametric elements. Ideally plastic material; (a), (b), (c) and (d) show various depths of indentation (reduced integration is used here).
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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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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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Page 87 and 88: 74 A general algorithm for compress
Page 105 and 106: 92 Incompressible laminar ¯ow Cons
Page 107 and 108: 94 Incompressible laminar ¯ow and,
Page 109 and 110: 96 Incompressible laminar ¯ow V/V
Page 111 and 112: 98 Incompressible laminar ¯ow The
Page 113 and 114: 100 Incompressible laminar ¯ow 1.5
Page 115 and 116: 102 Incompressible laminar ¯ow p 1
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Page 119 and 120: 106 Incompressible laminar ¯ow x 2
Page 121 and 122: 108 Incompressible laminar ¯ow Thi
Page 127 and 128: 114 Incompressible laminar ¯ow (a)
Page 129 and 130: 116 Incompressible laminar ¯ow (b)
Page 131 and 132: 118 Incompressible laminar ¯ow wit
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Page 141 and 142: 128 Incompressible laminar ¯ow C L
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Page 147 and 148: 134 Incompressible laminar ¯ow Fig
Page 149 and 150: 136 Incompressible laminar ¯ow 28.
Page 151 and 152: 138 Incompressible laminar ¯ow 71.
Page 153 and 154: 140 Incompressible laminar ¯ow 114
Page 155 and 156: 142 Incompressible laminar ¯ow 153
Page 157 and 158: 144 Free surfaces, buoyancy and tur
Page 183 and 184: 170 Compressible high-speed gas ¯o
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176 Compressible high-speed gas ¯o
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7 Shallow-water problems 7.1 Introd
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220 Shallow-water problems reduced
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222 Shallow-water problems Approxim
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224 Shallow-water problems These si
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226 Shallow-water problems h = 2 η
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228 Shallow-water problems Surface
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230 Shallow-water problems FL: 578
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232 Shallow-water problems B Fig. 7
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234 Shallow-water problems Time = 0
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236 Shallow-water problems Fig. 7.1
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238 Shallow-water problems where no
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240 Shallow-water problems 14. T.D.
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8 Waves Peter Bettess 8.1 Introduct
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244 Waves the familiar form where
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246 Waves of 10 nodes or thereabout
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248 Waves and the Astley wave envel
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250 Waves agreement with the analyt
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252 Waves Fig. 8.3 General wave dom
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254 Waves Relative errors (%) 10 8
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256 Waves (The indirect boundary in
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258 Waves where m is the number of
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260 Waves Fig. 17.6 of Volume 1. La
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262 Waves to using such coordinate
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264 Waves r/a 5 4 3 2 1 0 0 2 4 6 8
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266 Waves 8.16 Three-dimensional ef
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268 Waves wave elevations in shallo
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270 Waves integrals. Preliminary re
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272 Waves 39. R.J. Astley. FE mode
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9 Computer implementation of the CB
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276 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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