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

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Cylinder radius: a = 1 m, radius of the meshed region R/a = 7 The short-wave problem 249 Fig. 8.2 Short waves diffracted by a cylinder, modelled using special ®nite elements, Laghrouche and Bettess. 24
250 Waves agreement with the analytical series solution. In this problem ka ˆ 8 , ˆ 0:25a, radius of cylinder, a ˆ 1, and the mesh extends to r ˆ 7a. For a conventional radial ®nite element mesh, the requirement of 10 nodes per wavelength would lead to a mesh with 424 160 degrees of freedom. But in the results shown, with 36 directions per node and 252 nodes there are only 9072 degrees of freedom. The dramatic reduction in the number of variables merits further investigation and development of the method. The method still has a number of uncertainties regarding the conditioning of the system matrix and the stability of the technique and a signi®cant problem remains in the numerical cost of integrating the element matrix. 8.6 Waves in unbounded domains (exterior surface wave problems) Problems in this category include the di€raction and refraction of waves close to ®xed and ¯oating structures, the determination of wave forces and wave response for o€shore structures and vessels, and the determination of wave patterns adjacent to coastlines, open harbours and breakwaters. In electromagnetics there are scattering problems of the type already described, and in acoustics we have various noise problems. In the interior or ®nite part of the domain, ®nite elements, exactly as described in Sec. 8.2 can be used, but special procedures must be adopted for the part of the domain extending to in®nity. The main di culty is that the problem has no outer boundary. This necessitates the use of a radiation condition. Such a condition was introduced in Chapter 19 of Volume 1, as Eq. (19.18), for the case of a one-dimensional wave, or a normally incident plane wave in two or more dimensions. Work by Bayliss et al. 26;27 has developed a suitable radiation condition, in the form of an in®nite series of operators. The starting point is the representation of the outgoing wave in the form of an in®nite series. Each term in the series is then annihilated by using a boundary operator. The sequence of boundary operators thus constitutes the radiation condition. In addition there is a classical form of the boundary condition for periodic problems, given by Sommerfeld. A summary of all available radiation conditions is given in Table 8.1. 8.6.1 Background to wave problems The simplest type of exterior, or unbounded wave problem is that of some exciting device which sends out waves which do not return. This is termed the radiation problem. The next type of exterior wave problem is where we have a known incoming wave which encounters an object, is modi®ed and then again radiates away to in®nity. This case is known as the scattering problem, and is more complicated, in as much as we have to deal with both incident and radiated waves. Even when both waves are linear, this can lead to complications. Both the above cases can be complicated by wave refraction, where the wave speeds change, because of changes in the medium, for example changes in water depth. Usually this phenomenon leads to changes in the wave direction. Waves can also re¯ect from boundaries, both physical and computational.
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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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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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50 Convection dominated problems To
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52 Convection dominated problems C
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60 Convection dominated problems Ma
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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
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Page 239 and 240: 226 Shallow-water problems h = 2 η
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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
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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: 248 Waves and the Astley wave envel
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 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
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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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