11 September 2006 06:29:32 AM FREE_FEM_NAVIER_STOKES (C++ version): Finite element solution of the steady incompressible Navier Stokes equations on a triangulated region in 2 dimensions.' - nu * ( Uxx + Uyy ) + UUx + VUy + dPdx = F1(x,y) - nu * ( Vxx + Vyy ) + UVx + VVy + dPdy = F2(x,y) Ux + Vy = F3(x,y). Boundary conditions: U(x,y) = U_BC(x,y) V(x,y) = V_BC(x,y) P(x,y) = P_BC(x,y) The finite element method uses Taylor-Hood triangular elements which are linear for pressure and quadratic for velocity. Maximum number of Newton iterations IT_MAX = 1 The fluid viscosity NU = 1 Current status: * Code is starting to work. Node file is "nodes6.txt". Element file is "triangles6.txt". Number of nodes = 169 First 10 nodes Row: 1 2 Col 1 0 0 2 0.0833 0 3 0.1667 0 4 0.25 0 5 0.3333 0 6 0.4167 0 7 0.5 0 8 0.5833 0 9 0.6667 0 10 0.75 0 Element order = 6 Number of elements = 72 First 10 elements Row: 1 2 3 4 5 6 Col 1 1 27 3 14 15 2 2 29 3 27 16 15 28 3 3 29 5 16 17 4 4 31 5 29 18 17 30 5 5 31 7 18 19 6 6 33 7 31 20 19 32 7 7 33 9 20 21 8 8 35 9 33 22 21 34 9 9 35 11 22 23 10 10 37 11 35 24 23 36 Quadrature order = 3 Dirichlet boundary condition on pressure will be applied at node 1 Total number of variables is 387 Variable indices per node: Node U V P 1 1 2 3 2 4 5 -1 3 6 7 8 4 9 10 -1 5 11 12 13 6 14 15 -1 7 16 17 18 8 19 20 -1 9 21 22 23 10 24 25 -1 11 26 27 28 12 29 30 -1 13 31 32 33 14 34 35 -1 15 36 37 -1 16 38 39 -1 17 40 41 -1 18 42 43 -1 19 44 45 -1 20 46 47 -1 21 48 49 -1 22 50 51 -1 23 52 53 -1 24 54 55 -1 25 56 57 -1 26 58 59 -1 27 60 61 62 28 63 64 -1 29 65 66 67 30 68 69 -1 31 70 71 72 32 73 74 -1 33 75 76 77 34 78 79 -1 35 80 81 82 36 83 84 -1 37 85 86 87 38 88 89 -1 39 90 91 92 40 93 94 -1 41 95 96 -1 42 97 98 -1 43 99 100 -1 44 101 102 -1 45 103 104 -1 46 105 106 -1 47 107 108 -1 48 109 110 -1 49 111 112 -1 50 113 114 -1 51 115 116 -1 52 117 118 -1 53 119 120 121 54 122 123 -1 55 124 125 126 56 127 128 -1 57 129 130 131 58 132 133 -1 59 134 135 136 60 137 138 -1 61 139 140 141 62 142 143 -1 63 144 145 146 64 147 148 -1 65 149 150 151 66 152 153 -1 67 154 155 -1 68 156 157 -1 69 158 159 -1 70 160 161 -1 71 162 163 -1 72 164 165 -1 73 166 167 -1 74 168 169 -1 75 170 171 -1 76 172 173 -1 77 174 175 -1 78 176 177 -1 79 178 179 180 80 181 182 -1 81 183 184 185 82 186 187 -1 83 188 189 190 84 191 192 -1 85 193 194 195 86 196 197 -1 87 198 199 200 88 201 202 -1 89 203 204 205 90 206 207 -1 91 208 209 210 92 211 212 -1 93 213 214 -1 94 215 216 -1 95 217 218 -1 96 219 220 -1 97 221 222 -1 98 223 224 -1 99 225 226 -1 100 227 228 -1 101 229 230 -1 102 231 232 -1 103 233 234 -1 104 235 236 -1 105 237 238 239 106 240 241 -1 107 242 243 244 108 245 246 -1 109 247 248 249 110 250 251 -1 111 252 253 254 112 255 256 -1 113 257 258 259 114 260 261 -1 115 262 263 264 116 265 266 -1 117 267 268 269 118 270 271 -1 119 272 273 -1 120 274 275 -1 121 276 277 -1 122 278 279 -1 123 280 281 -1 124 282 283 -1 125 284 285 -1 126 286 287 -1 127 288 289 -1 128 290 291 -1 129 292 293 -1 130 294 295 -1 131 296 297 298 132 299 300 -1 133 301 302 303 134 304 305 -1 135 306 307 308 136 309 310 -1 137 311 312 313 138 314 315 -1 139 316 317 318 140 319 320 -1 141 321 322 323 142 324 325 -1 143 326 327 328 144 329 330 -1 145 331 332 -1 146 333 334 -1 147 335 336 -1 148 337 338 -1 149 339 340 -1 150 341 342 -1 151 343 344 -1 152 345 346 -1 153 347 348 -1 154 349 350 -1 155 351 352 -1 156 353 354 -1 157 355 356 357 158 358 359 -1 159 360 361 362 160 363 364 -1 161 365 366 367 162 368 369 -1 163 370 371 372 164 373 374 -1 165 375 376 377 166 378 379 -1 167 380 381 382 168 383 384 -1 169 385 386 387 The matrix half bandwidth is 61 The matrix bandwidth is 123 The storage bandwidth is 184 Solution to the STOKES Equations: Node U V P 0 0 0 0 1 -1.04208e-17 -1.56313e-17 2 1.56031e-17 -1.04021e-17 1.48682 3 -5.21043e-18 -5.21042e-18 4 -2.59974e-17 -2.07979e-17 1.40833 5 -5.71772e-17 -4.32658e-33 6 7.80156e-17 1.04021e-17 1.86221 7 1.04209e-17 -8.6823e-34 8 6.24125e-17 -2.08042e-17 2.34993 9 2.08292e-17 2.38405e-32 10 -1.0399e-17 2.07979e-17 2.42598 11 -1.03959e-17 -3.11876e-17 12 0 0 2.26001 13 -0 -0 14 -0.00984613 -0.00541645 15 -0.0112559 0.00576421 16 -0.0230026 0.00767544 17 -0.0352985 0.00550562 18 -0.0429834 0.0030503 19 -0.0455948 -0.000257255 20 -0.0428229 -0.00333781 21 -0.0336784 -0.00561425 22 -0.0225337 -0.00671657 23 -0.0112583 -0.00532941 24 -0.00234559 -0.00204037 25 0 0 26 0 0 1.86702 27 -0.00591988 0.0106333 28 -0.0226039 0.0191649 1.44632 29 -0.0441484 0.0223465 30 -0.0629404 0.0188154 1.58791 31 -0.0756665 0.0109938 32 -0.0804996 -0.00019139 1.87559 33 -0.0754115 -0.010647 34 -0.0615359 -0.0193257 2.18474 35 -0.0423621 -0.0231173 36 -0.0223635 -0.0200727 2.24664 37 -0.00629978 -0.0115643 38 0 0 2.02127 39 0 -5.98723e-17 40 -0.00879239 0.0256702 41 -0.0328731 0.0418145 42 -0.0621612 0.0461211 43 -0.0869426 0.0381242 44 -0.104232 0.0209444 45 -0.110129 -0.000986154 46 -0.103879 -0.0225111 47 -0.0858971 -0.0396753 48 -0.0604325 -0.047478 49 -0.0332093 -0.0431512 50 -0.00992836 -0.0258664 51 0 0 52 0 0 1.75552 53 -0.0129432 0.0465599 54 -0.0427499 0.0736417 1.38194 55 -0.0791124 0.0791889 56 -0.109912 0.0632721 1.50773 57 -0.129856 0.0347093 58 -0.136914 -0.000817623 1.90913 59 -0.129245 -0.0354393 60 -0.10825 -0.0641953 2.30966 61 -0.0770006 -0.0780825 62 -0.0428187 -0.0726148 2.40434 63 -0.0133117 -0.0459153 64 0 0 2.00175 65 0 0 66 -0.0158841 0.0731168 67 -0.0516634 0.112338 68 -0.0964482 0.115293 69 -0.130609 0.0905025 70 -0.151729 0.048691 71 -0.158811 -0.00146305 72 -0.151361 -0.0515707 73 -0.12821 -0.0932884 74 -0.0926889 -0.115281 75 -0.0529691 -0.110975 76 -0.0174183 -0.0726627 77 0 0 78 0 8.84002e-17 1.14842 79 -0.0188229 0.104065 80 -0.060942 0.153409 0.569976 81 -0.10997 0.154517 82 -0.144199 0.119286 1.13889 83 -0.163023 0.0635383 84 -0.170289 -0.00177415 1.91902 85 -0.163008 -0.0663498 86 -0.140927 -0.122815 2.77081 87 -0.104977 -0.155965 88 -0.0617041 -0.156833 3.03783 89 -0.019396 -0.106617 90 0 0 2.67218 91 0 0 92 -0.022668 0.14334 93 -0.0677864 0.200375 94 -0.114395 0.190743 95 -0.14074 0.142543 96 -0.153032 0.0746418 97 -0.157487 -0.00174914 98 -0.153628 -0.0782995 99 -0.138422 -0.147503 100 -0.113288 -0.198285 101 -0.0703859 -0.208103 102 -0.0137877 -0.138057 103 0 0 104 0 2.46683e-16 -0.277326 105 -0.0298615 0.207741 106 -0.0750249 0.251625 -0.937195 107 -0.0975379 0.220618 108 -0.10556 0.158347 0.484682 109 -0.103136 0.0823025 110 -0.103471 0.00142283 1.99119 111 -0.104376 -0.0788458 112 -0.106851 -0.156245 3.59826 113 -0.100474 -0.219339 114 -0.0669997 -0.24944 5.19932 115 -0.01464 -0.181723 116 0 0 3.20421 117 0 0 118 -0.0494694 0.238565 119 -0.0694256 0.259735 120 -0.0361025 0.210976 121 -0.00765 0.14332 122 0.0120159 0.0725679 123 0.0190523 0.00203387 124 0.0131853 -0.0676572 125 -0.00980157 -0.139227 126 -0.0481527 -0.218754 127 -0.0531942 -0.272788 128 -0.06092 -0.256306 129 0 0 130 0 1.06468e-16 -8.27472 131 -0.0168331 0.262765 132 0.00793891 0.223465 -3.68775 133 0.122651 0.138895 134 0.193126 0.0833036 -0.670392 135 0.224548 0.036433 136 0.241086 -0.00248779 1.88702 137 0.226496 -0.0470371 138 0.181587 -0.0946033 4.06982 139 0.100056 -0.161507 140 0.0176085 -0.222141 8.67272 141 -0.0542776 -0.221212 142 0 0 16.5092 143 0 0 144 0.165459 0.170105 145 0.267422 0.0925736 146 0.451842 0.0502135 147 0.526474 0.0259181 148 0.559088 0.0103283 149 0.571536 -0.001389 150 0.56168 -0.0141363 151 0.516587 -0.031053 152 0.441663 -0.0605785 153 0.339303 -0.0967744 154 0.1532 -0.153799 155 0 0 156 1 0 -25.6472 157 1 0 158 1 0 -1.93648 159 1 0 160 1 0 0.289823 161 1 0 162 1 0 1.86864 163 1 0 164 1 0 2.99201 165 1 0 166 1 0 7.50014 167 1 0 168 1 0 31.9395 Pressure nodes written to "nodes3.txt". Pressure triangles written to "triangles3.txt". Stokes pressures written to "stokes_pressure3.txt". Stokes velocities written to "stokes_velocity6.txt". l2-norm of FEM residual = 3.67239 l2-norm of adjusted FEM residual = 0.0302317 l2-norm of Newton correction = 0.379819 l2-norm of FEM residual = 3.67143 l2-norm of adjusted FEM residual = 6.78595e-07 Convergence. Navier Stokes pressures written to "navier_stokes_pressure3.txt". Navier Stokes velocities written to "navier_stokes_velocity6.txt". FREE_FEM_STOKES: Normal end of execution. 11 September 2006 06:29:32 AM