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

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Inputs Preliminary routines Passed Data check Step 1 Intermediate momentum Step 2 Density/pressure Step 3 Momentum correction Energy coupling No Boundary conditions Steady state Yes Output End Fig. 9.2 Flow diagram for CBS¯ow. Failed Yes No Make changes Energy/temperature calculation Time loop Solution module 279
280 Computer implementation of the CBS algorithm 9.3.2 Time step In general, three di€erent ways of establishing the time steps are possible. In problems where only the steady state is of importance, so-called `local time stepping' is used (see Sec. 3.3.4, Chapter 3). Here a local time step at each and every nodal points is calculated and used in the computation. When we seek accurate transient solution of any problem, the so-called `minimum step' value is used. Here the minimum of all local time step values is calculated and used in the computation. Another and less frequently used option is that of giving a `®xed' user-prescribed time step value. Selection of such a quantity needs considerable experience from solving several ¯ow problems. The times loop starts with a subroutine where the above-mentioned time step options are available. In general the local time steps are calculated at every iteration for the initial few time steps and then they are calculated only after a certain number of iterations as prescribed by the user. If the last option of the user-speci®ed ®xed time step is used, the local time steps are not calculated. Figure 9.3 shows the subroutine used for calculating the local time steps for inviscid compressible ¯ows with linear triangular elements. As indicated in Sec. 4.3.3, Chapter 4, two di€erent time steps are often useful in getting better stabilization procedures. 4 Such internal (DELTI) and external (DELTP) time stepping options are available in the routine of Fig. 9.3. 9.3.3 Shockcapture The CBS algorithm introduces naturally some terms to stabilize the oscillations generated by the convective acceleration. However, for compressible high-speed ¯ows, these terms are not su cient to suppress the oscillations in the vicinity of shocks and some additional arti®cial viscosity terms need to be added (see Sec. 6.5, Chapter 6). We have given two di€erent forms of arti®cial viscosities based on the second derivative of pressure in the program. Another possibility is to use anisotropic shock capturing based on the residual of individual equations solved. However we have not used the second alternative in the program as the second derivative based procedures give quite satisfactory results for all high-speed ¯ow problems. In the ®rst method implemented, we need to calculate a pressure switch (see Eq. (6.16), Chapter 6) from the nodal pressure values. Figure 9.4 gives a typical example of triangular elements inside and on the boundaries. For inside nodes (Fig. 9.4(a)) we calculate the nodal switch as S 1 ˆ j4p 1 ÿ p 2 ÿ p 3 ÿ p 4 ÿ p 5j jp 1 ÿ p 2j‡jp 1 ÿ p 3j‡jp 1 ÿ p 4j‡jp 1 ÿ p 5j and for the boundary node (Fig. 9.4(b)) we calculate S 1 ˆ j5p 1 ÿ 2p 2 ÿ p 3 ÿ 2p 4j 2jp 1 ÿ p 2j‡jp 1 ÿ p 3j‡2jp 1 ÿ p 4j …9:1† …9:2†
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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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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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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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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
Page 251 and 252: 238 Shallow-water problems where no
Page 253 and 254: 240 Shallow-water problems 14. T.D.
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
Page 287 and 288: 9 Computer implementation of the CB
Page 289 and 290: 276 Computer implementation of the
Page 291: 278 Computer implementation of the
Page 305 and 306: 292 Appendix A Again substituting t
Page 307 and 308: 294 Appendix B As usual the domain
Page 309 and 310: 296 Appendix B 5. While the piecewi
Page 311 and 312: Appendix C Edge-based ®nite elemen
Page 313 and 314: Appendix D Multigrid methods It is
Page 315 and 316: Appendix E Boundary layer±inviscid
Page 317 and 318: 304 Appendix E In Fig. E.2, Cp is t
Page 320 and 321: Author index Page numbers in bold r
Page 322 and 323: Gerdes, K. 259, 264, 265, 272 Ghia,
Page 324 and 325: Nakos, D.E. 146, 165 Narayana, P.A.
Page 326: Wu, J. 67, 90; 102, 108, 112, 137;
Page 329 and 330: 316 Subject index Blurring of the s
Page 331 and 332: 318 Subject index Convected coordin
Page 333 and 334: 320 Subject index Elements ± cont.
Page 335 and 336: 322 Subject index Flows: buoyancy d
Page 337 and 338: 324 Subject index Incompressible St
Page 339 and 340: 326 Subject index Metals: hot, 120
Page 341 and 342: 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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