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

More documents

Recommendations

Info

Wave transmission coefficient C T Incident wave 1.0 0.5 Eigenfunction method Present method (twodimensional model) 0 0 1 2 3 4 5 6 7 8 9 10 Wave period T (s) 0.1 0.5 1.0 2.0 3.0 4.0 5.0 6.0 6.0 7.0 8.0 Wavelength λ/B 3m Infinite elements Special threedimensional finite elements Three-dimensional finite elements Incident wave 1.2 1.4 10 m Floating breakwater 0.8 1.0 0.6 0.4 0.6 0.8 1.0 1.0 1.2 1.4 1.6 1.8 Floating breakwater 2.0 2.2 λ = 34.9 m Three-dimensional effects in surface waves 267 η 1 = 1 m Bed 0.8 0.6 0.4 0.2 20 m Free surface 8m Unit: m Fig. 8.11 Element mesh, contours of wave elevation and wave transmission coef®cients for ¯oating breakwater, Hara. 67
268 Waves wave elevations in shallow water and large wave elevations in intermediate to deep water. We have discussed a similar problem in Chapter 5. 8.16.2 Cnoidal and solitary waves The equations modelled in Chapter 7 can deal with large-amplitude waves in shallow water. These are called cnoidal waves when periodic, and solitary waves when the period is in®nite. For more details see references 1±4. The ®nite element methodology of Chapter 7, can be used to model the propagation of such waves. It is also possible to reduce the equations of momentum balance and mass conservation to corresponding wave equations in one variable, of which there are several di€erent forms. One famous equation is the Korteweg±de Vries equation, which in physical variables is @ p ‡ gH @t 1 ‡ 3 2h @ @x ‡ h2 6 p @ gH 3 ˆ 0 …8:46† 3 @x This equation has been given a great deal of attention by mathematicians. It can be solved directly using ®nite element methods, and a general introduction to this ®eld is given by Mitchell and Schoombie. 68 8.16.3 Stokes waves When the water is deep, a di€erent asymptotic expansion can be used in which the velocity potential, , and the surface elevation, , are expanded in terms of a small parameter, ", which can be identi®ed with the slope of the water surface. When these expressions are substituted into the free surface condition, and terms with the same order in " are collected, a series of free surface conditions is obtained. The equations were solved by Stokes initially, and then by other workers, to very high orders, to give solutions for large-amplitude progressive waves in deep water. There is an extensive literature on these solutions, and they are used in the o€shore industry for calculating loads on o€shore structures. In recent years, attempts have been made to model the second-order wave di€raction problem, using ®nite elements, and similar techniques. The ®rst-order di€raction problem is as described in Sec. 8.9. In the second-order problem, the free surface condition now involves the ®rst-order potential. First order: Second order: @ …1† @z @ …2† @z ÿ !2 g ÿ !2 g …1† ˆ 0 …8:47† …2† ˆ …2† D …8:48†
Page 1 and 2:
The Finite Element Method Fifth edi
Page 3 and 4:
The Finite Element Method Fifth edi
Page 5 and 6:
Dedication This book is dedicated t
Page 7 and 8:
3.7 The performance of two- and sin
Page 9 and 10:
...................................
Page 11 and 12:
Volume 2: Solid and structural mech
Page 13 and 14:
xiv Preface to Volume 3 this algori
Page 15 and 16:
2 Introduction and the equations of
Page 17 and 18:
Page 19 and 20:
Page 21 and 22:
Page 23 and 24:
10 Introduction and the equations o
Page 25 and 26:
12 Introduction and the equations o
Page 27 and 28:
14 Convection dominated problems We
Page 29 and 30:
16 Convection dominated problems wh
Page 31 and 32:
18 Convection dominated problems ch
Page 33 and 34:
20 Convection dominated problems i.
Page 35 and 36:
22 Convection dominated problems 2
Page 37 and 38:
24 Convection dominated problems 2.
Page 39 and 40:
26 Convection dominated problems ar
Page 41 and 42:
28 Convection dominated problems Th
Page 43 and 44:
30 Convection dominated problems co
Page 45 and 46:
32 Convection dominated problems 2.
Page 47 and 48:
34 Convection dominated problems ha
Page 49 and 50:
36 Convection dominated problems an
Page 51 and 52:
38 Convection dominated problems ac
Page 53 and 54:
40 Convection dominated problems An
Page 55 and 56:
42 Convection dominated problems U
Page 57 and 58:
44 Convection dominated problems (a
Page 59 and 60:
46 Convection dominated problems sp
Page 61 and 62:
48 Convection dominated problems ag
Page 63 and 64:
50 Convection dominated problems To
Page 65 and 66:
52 Convection dominated problems C
Page 67 and 68:
Page 69 and 70:
56 Convection dominated problems an
Page 71 and 72:
Page 73 and 74:
60 Convection dominated problems Ma
Page 75 and 76:
62 Convection dominated problems 47
Page 77 and 78:
3 A general algorithm for compressi
Page 79 and 80:
66 A general algorithm for compress
Page 81 and 82:
Page 83 and 84:
Page 85 and 86:
Page 87 and 88:
Page 89 and 90:
Page 91 and 92:
Page 93 and 94:
Page 95 and 96:
Page 97 and 98:
Page 99 and 100:
Page 101 and 102:
Page 103 and 104:
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
Page 117 and 118:
104 Incompressible laminar ¯ow �
Page 119 and 120:
106 Incompressible laminar ¯ow x 2
Page 121 and 122:
108 Incompressible laminar ¯ow Thi
Page 123 and 124:
Page 125 and 126:
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
Page 133 and 134:
120 Incompressible laminar ¯ow The
Page 135 and 136:
122 Incompressible laminar ¯ow Fig
Page 137 and 138:
124 Incompressible laminar ¯ow and
Page 139 and 140:
126 Incompressible laminar ¯ow U U
Page 141 and 142:
128 Incompressible laminar ¯ow C L
Page 143 and 144:
130 Incompressible laminar ¯ow are
Page 145 and 146:
132 Incompressible laminar ¯ow sha
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 159 and 160:
Page 161 and 162:
Page 163 and 164:
Page 165 and 166:
Page 167 and 168:
Page 169 and 170:
Page 171 and 172:
Page 173 and 174:
Page 175 and 176:
Page 177 and 178:
Page 179 and 180:
Page 181 and 182:
Page 183 and 184:
170 Compressible high-speed gas ¯o
Page 185 and 186:
Page 187 and 188:
Page 189 and 190:
Page 191 and 192:
Page 193 and 194:
Page 195 and 196:
Page 197 and 198:
Page 199 and 200:
Page 201 and 202:
Page 203 and 204:
Page 205 and 206:
Page 207 and 208:
Page 209 and 210:
Page 211 and 212:
Page 213 and 214:
Page 215 and 216:
Page 217 and 218:
Page 219 and 220:
Page 221 and 222:
Page 223 and 224:
Page 225 and 226:
Page 227 and 228:
Page 229 and 230: 216 Compressible high-speed gas ¯o
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
Page 237 and 238: 224 Shallow-water problems These si
Page 239 and 240: 226 Shallow-water problems h = 2 η
Page 241 and 242: 228 Shallow-water problems Surface
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
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: 266 Waves 8.16 Three-dimensional ef
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
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
Page 343 and 344:
330 Subject index Shock interaction
Page 345 and 346:
332 Subject index Theorem ± cont.
Page 347:
334 Subject index Wave ± cont. ste
show all

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

Create successful ePaper yourself

Delete template?

Save as template?