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

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Three-dimensional inviscid examples in steady state 189 and the process is repeated on several meshes done subsequently. This procedure can be used on several meshes and the iterative process is much accelerated. We discuss this process in Appendix D in a little more detail. However, we quote here several references 65ÿ69 in which such multigrid procedures have been used and these are of considerable value. Multigrid methods are obviously designed for meshes which are `nested' i.e. in which coarser and ®ner mesh nodes coincide. This need not be the case generally. In many applications completely di€erent meshes of varying density are used at various stages. Parallel computation The third procedure of reducing the solution time is to use parallization. We do not discuss it here in detail as the matter is potentially coupled with the computational aspects of the problem. Here the reader should consult the current literature on the subject. 36ÿ38 In what follows we shall illustrate three-dimensional applications on a few inviscid examples as this section deals with Euler problems. However in Sec. 6.10 we shall return to a fully three-dimensional formulation using viscous Navier±Stokes equations. 6.8.1 Solution of the ¯ow pattern around a complete aircraft In the early days of numerical analysis applied to computational ¯uid dynamics which used ®nite di€erences, no complete aircraft was analysed as in general only structured meshes were admissible. The analysis thus had to be carried out on isolated components of the aircraft. Later construction of distorted and partly structured meshes increased the possibility of analysis. Nevertheless the ®rst complete aircraft analyses were done only in the mid-1980s. In all of these, ®nite elements using unstructured meshes were used (though we include here the ®nite volume formulation which was almost identical to ®nite elements and was used by Jameson et al. 70 ). The very ®rst aircraft was the one dealt with using potential theory in the Dassault establishment. The results were publisshed later by Periaux and coworkers. 71 Very shortly after that a complete supersonic aircraft was analysed by Peraire et al. 16 in Swansea in 1987. Figure 6.12 shows the aircraft analysed in Swansea 16 which is a supersonic ®ghter of generic type at Mach 2. The analysis was made slightly adaptive though adaptivity was not carried very far (due to cost). Nevertheless the re®nement localized the shocks which formed. In the analysis some 125 000 elements were used with approximately 70 000 nodes and therefore some 350 000 variables. This of course is not a precise analysis and many more variables would be used currently to get a more accurate representation of ¯ow and pressure variables. A more sophisticated analysis is shown in the plate at the front of the book. Here a civil aircraft in subsonic ¯ow is modelled and this illustrates the use of multigrid
190 Compressible high-speed gas ¯ow Fig. 6.12 Adaptive three-dimensional solution of compressible inviscid ¯ow around a high speed (Mach 2) aircraft. 16 Nodes: 70 000, elements: 125 000. methods. In this particular multigrid applications three meshes of di€erent re®nement were used and the iteration is fully described in reference 26. In this example the total number of unknown quantities was 1 616 000 in the ®nest mesh and indeed the details of the subdivision are given in the legend of the plate.
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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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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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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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Page 231 and 232: 7 Shallow-water problems 7.1 Introd
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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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Zienkiewicz O.C., Taylor R.L. Vol. 3. The finite - tiera.ru

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