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

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steady state was reached after some 500 time steps. For the case with supersonic in¯ow and subsonic out¯ow, a shock forms and Lapidus-type arti®cial di€usion was used to deal with it, showing in Fig. 6.3(c) the increasing amount of `smearing' as the coe cient C Lap is increased. 1.0 Inflow 0.6 (a) Structured uniform mesh t = 0.5 t = 1.5 t = 2.5 t = 4.0 y 3.0 (b) Solution – contours of pressure at various times Some preliminary examples for the Euler equation 179 x 2016 elements 1089 nodes Fig. 6.4 Transient supersonic ¯ow over a step in a wind tunnel 4 (problem of Woodward and Colella 59 ). In¯ow Mach 3 uniform ¯ow.
180 Compressible high-speed gas ¯ow 6.6.3 Two-dimensional transient supersonic ¯ow over a step This ®nal example concerns the transient initiation of supersonic ¯ow in a wind tunnel containing a step. The problem was ®rst studied by Woodward and Colella 59 and the results of reference 4 presented here are essentially similar. In this problem a uniform mesh of linear triangles, shown in Fig. 6.4, was used and no di culties of computation were encountered although a Lapidus constant C Lap ˆ 2:0 had to be used due to the presence of shocks. 6.7 Adaptive re®nement and shock capture in Euler problems 6.7.1 General The examples of the previous section have indicated the formation of shocks both in transient and steady-state problems of high-speed ¯ow. Clearly the resolution of such discontinuities or near discontinuities requires a very ®ne mesh. Here the use of `engineering judgement', which is often used in solid mechanics by designing a priori mesh re®ning near singularities posed by corners in the boundary, etc., can no longer be used. In problems of compressible ¯ow the position of shocks, where the re®nement is most needed, is not known in advance. For this and other reasons, the use of adaptive mesh re®nement based on error indicators is essential for obtaining good accuracy and `capturing' the location of shocks. It is therefore not surprising that the science of adaptive re®nement has progressed rapidly in this area and indeed, as we shall see later, has been extended to deal with Navier±Stokes equations where a higher degree of re®nement is also required in boundary layers. We have discussed the history of such adaptive development and procedures for its use in Sec. 4.5, Chapter 4. 6.7.2 The h-re®nement process and mesh enrichment Once an approximate solution has been achieved on a given mesh, the local errors can be evaluated and new element sizes (and elongation directions if used) can be determined for each element. For some purposes it is again convenient to transfer such values to the nodes so that they can be interpolated continuously. The procedure here is of course identical to that of smoothing the derivatives discussed in Sec. 4.5, Chapter 4. To achieve the desired accuracy various procedures can be used. The most obvious is the process of mesh enrichment in which the existing mesh is locally subdivided into smaller elements still retaining the `old' mesh in the con®guration. Figure 6.5(a) shows how triangles can be readily subdivided in this way. With such enrichment an obvious connectivity di culty appears. This concerns the manner in which the subdivided
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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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...................................
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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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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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Page 231 and 232: 7 Shallow-water problems 7.1 Introd
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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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