October 7 2006 3:09:40.714 PM FREE_FEM_NAVIER_STOKES (FORTRAN90 version): Finite element solution of the the steady incompressible Navier Stokes equations on a triangulated grid 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 may be of Dirichlet type: U(x,y) = U_BC(x,y) V(x,y) = V_BC(x,y) P(x,y) = P_BC(x,y) or of Neumann type with zero right hand side: dU/dn(x,y) = 0 dV/dn(x,y) = 0 dP/dn(x,y) = 0 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 value NU = 1.00000 Current status: * testing zero Neumann condition option. Node file is "nodes6.txt". Element file is "triangles6.txt". Number of nodes = 169 First 10 nodes Row 1 2 Col 1 0.00000 0.00000 2 0.833000E-01 0.00000 3 0.166700 0.00000 4 0.250000 0.00000 5 0.333300 0.00000 6 0.416700 0.00000 7 0.500000 0.00000 8 0.583300 0.00000 9 0.666700 0.00000 10 0.750000 0.00000 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 Number of Neumann conditions added = 0 Boundary conditions per node: Node U_cond V_cond P_cond 1 2 2 2 2 2 2 0 3 2 2 1 4 2 2 0 5 2 2 1 6 2 2 0 7 2 2 1 8 2 2 0 9 2 2 1 10 2 2 0 11 2 2 1 12 2 2 0 13 2 2 1 14 2 2 0 15 1 1 0 16 1 1 0 17 1 1 0 18 1 1 0 19 1 1 0 20 1 1 0 21 1 1 0 22 1 1 0 23 1 1 0 24 1 1 0 25 1 1 0 26 2 2 0 27 2 2 1 28 1 1 0 29 1 1 1 30 1 1 0 31 1 1 1 32 1 1 0 33 1 1 1 34 1 1 0 35 1 1 1 36 1 1 0 37 1 1 1 38 1 1 0 39 2 2 1 40 2 2 0 41 1 1 0 42 1 1 0 43 1 1 0 44 1 1 0 45 1 1 0 46 1 1 0 47 1 1 0 48 1 1 0 49 1 1 0 50 1 1 0 51 1 1 0 52 2 2 0 53 2 2 1 54 1 1 0 55 1 1 1 56 1 1 0 57 1 1 1 58 1 1 0 59 1 1 1 60 1 1 0 61 1 1 1 62 1 1 0 63 1 1 1 64 1 1 0 65 2 2 1 66 2 2 0 67 1 1 0 68 1 1 0 69 1 1 0 70 1 1 0 71 1 1 0 72 1 1 0 73 1 1 0 74 1 1 0 75 1 1 0 76 1 1 0 77 1 1 0 78 2 2 0 79 2 2 1 80 1 1 0 81 1 1 1 82 1 1 0 83 1 1 1 84 1 1 0 85 1 1 1 86 1 1 0 87 1 1 1 88 1 1 0 89 1 1 1 90 1 1 0 91 2 2 1 92 2 2 0 93 1 1 0 94 1 1 0 95 1 1 0 96 1 1 0 97 1 1 0 98 1 1 0 99 1 1 0 100 1 1 0 101 1 1 0 102 1 1 0 103 1 1 0 104 2 2 0 105 2 2 1 106 1 1 0 107 1 1 1 108 1 1 0 109 1 1 1 110 1 1 0 111 1 1 1 112 1 1 0 113 1 1 1 114 1 1 0 115 1 1 1 116 1 1 0 117 2 2 1 118 2 2 0 119 1 1 0 120 1 1 0 121 1 1 0 122 1 1 0 123 1 1 0 124 1 1 0 125 1 1 0 126 1 1 0 127 1 1 0 128 1 1 0 129 1 1 0 130 2 2 0 131 2 2 1 132 1 1 0 133 1 1 1 134 1 1 0 135 1 1 1 136 1 1 0 137 1 1 1 138 1 1 0 139 1 1 1 140 1 1 0 141 1 1 1 142 1 1 0 143 2 2 1 144 2 2 0 145 1 1 0 146 1 1 0 147 1 1 0 148 1 1 0 149 1 1 0 150 1 1 0 151 1 1 0 152 1 1 0 153 1 1 0 154 1 1 0 155 1 1 0 156 2 2 0 157 2 2 1 158 2 2 0 159 2 2 1 160 2 2 0 161 2 2 1 162 2 2 0 163 2 2 1 164 2 2 0 165 2 2 1 166 2 2 0 167 2 2 1 168 2 2 0 169 2 2 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 1 0.00000 0.00000 0.00000 2 0.521042E-17 0.104208E-16 3 0.156031E-16 -0.104021E-16 1.48682 4 -0.260334E-16 0.521042E-17 5 0.259974E-16 0.207979E-16 1.40833 6 -0.259896E-16 -0.519793E-17 7 0.520104E-16 0.00000 1.86221 8 -0.208167E-16 0.208667E-16 9 -0.208042E-16 0.00000 2.34993 10 0.104146E-16 0.260079E-31 11 -0.103990E-16 0.207979E-16 2.42598 12 -0.423904E-31 -0.207917E-16 13 0.00000 0.00000 2.26001 14 -0.693889E-17 -0.138778E-16 15 -0.984613E-02 -0.541645E-02 16 -0.112559E-01 0.576421E-02 17 -0.230026E-01 0.767544E-02 18 -0.352985E-01 0.550562E-02 19 -0.429834E-01 0.305030E-02 20 -0.455948E-01 -0.257255E-03 21 -0.428229E-01 -0.333781E-02 22 -0.336784E-01 -0.561425E-02 23 -0.225337E-01 -0.671657E-02 24 -0.112583E-01 -0.532941E-02 25 -0.234559E-02 -0.204037E-02 26 0.00000 0.00000 27 -0.208042E-16 0.00000 1.86702 28 -0.591988E-02 0.106333E-01 29 -0.226039E-01 0.191649E-01 1.44632 30 -0.441484E-01 0.223465E-01 31 -0.629404E-01 0.188154E-01 1.58791 32 -0.756665E-01 0.109938E-01 33 -0.804996E-01 -0.191390E-03 1.87559 34 -0.754115E-01 -0.106470E-01 35 -0.615359E-01 -0.193257E-01 2.18474 36 -0.423621E-01 -0.231173E-01 37 -0.223635E-01 -0.200727E-01 2.24664 38 -0.629978E-02 -0.115643E-01 39 0.00000 0.00000 2.02127 40 0.00000 0.468566E-16 41 -0.879239E-02 0.256702E-01 42 -0.328731E-01 0.418145E-01 43 -0.621612E-01 0.461211E-01 44 -0.869426E-01 0.381242E-01 45 -0.104232 0.209444E-01 46 -0.110129 -0.986154E-03 47 -0.103879 -0.225111E-01 48 -0.858971E-01 -0.396753E-01 49 -0.604325E-01 -0.474780E-01 50 -0.332093E-01 -0.431512E-01 51 -0.992836E-02 -0.258664E-01 52 0.00000 0.00000 53 0.00000 0.00000 1.75552 54 -0.129432E-01 0.465599E-01 55 -0.427499E-01 0.736417E-01 1.38194 56 -0.791124E-01 0.791889E-01 57 -0.109912 0.632721E-01 1.50773 58 -0.129856 0.347093E-01 59 -0.136914 -0.817623E-03 1.90913 60 -0.129245 -0.354393E-01 61 -0.108250 -0.641953E-01 2.30966 62 -0.770006E-01 -0.780825E-01 63 -0.428187E-01 -0.726148E-01 2.40434 64 -0.133117E-01 -0.459153E-01 65 0.00000 0.00000 2.00175 66 0.00000 0.00000 67 -0.158841E-01 0.731168E-01 68 -0.516634E-01 0.112338 69 -0.964482E-01 0.115293 70 -0.130609 0.905025E-01 71 -0.151729 0.486910E-01 72 -0.158811 -0.146305E-02 73 -0.151361 -0.515707E-01 74 -0.128210 -0.932884E-01 75 -0.926889E-01 -0.115281 76 -0.529691E-01 -0.110975 77 -0.174183E-01 -0.726627E-01 78 0.00000 0.00000 79 0.00000 -0.116755E-16 1.14842 80 -0.188229E-01 0.104065 81 -0.609420E-01 0.153409 0.569976 82 -0.109970 0.154517 83 -0.144199 0.119286 1.13889 84 -0.163023 0.635383E-01 85 -0.170289 -0.177415E-02 1.91902 86 -0.163008 -0.663498E-01 87 -0.140927 -0.122815 2.77081 88 -0.104977 -0.155965 89 -0.617041E-01 -0.156833 3.03783 90 -0.193960E-01 -0.106617 91 0.00000 0.00000 2.67218 92 0.00000 0.00000 93 -0.226680E-01 0.143340 94 -0.677864E-01 0.200375 95 -0.114395 0.190743 96 -0.140740 0.142543 97 -0.153032 0.746418E-01 98 -0.157487 -0.174914E-02 99 -0.153628 -0.782995E-01 100 -0.138422 -0.147503 101 -0.113288 -0.198285 102 -0.703859E-01 -0.208103 103 -0.137877E-01 -0.138057 104 0.00000 0.00000 105 0.00000 -0.216758E-15 -0.277326 106 -0.298615E-01 0.207741 107 -0.750249E-01 0.251625 -0.937195 108 -0.975379E-01 0.220618 109 -0.105560 0.158347 0.484682 110 -0.103136 0.823025E-01 111 -0.103471 0.142283E-02 1.99119 112 -0.104376 -0.788458E-01 113 -0.106851 -0.156245 3.59826 114 -0.100474 -0.219339 115 -0.669997E-01 -0.249440 5.19932 116 -0.146400E-01 -0.181723 117 0.00000 0.00000 3.20421 118 0.00000 0.00000 119 -0.494694E-01 0.238565 120 -0.694256E-01 0.259735 121 -0.361025E-01 0.210976 122 -0.765000E-02 0.143320 123 0.120159E-01 0.725679E-01 124 0.190523E-01 0.203387E-02 125 0.131853E-01 -0.676572E-01 126 -0.980157E-02 -0.139227 127 -0.481527E-01 -0.218754 128 -0.531942E-01 -0.272788 129 -0.609200E-01 -0.256306 130 0.00000 0.00000 131 0.00000 -0.279479E-15 -8.27472 132 -0.168331E-01 0.262765 133 0.793891E-02 0.223465 -3.68775 134 0.122651 0.138895 135 0.193126 0.833036E-01 -0.670392 136 0.224548 0.364330E-01 137 0.241086 -0.248779E-02 1.88702 138 0.226496 -0.470371E-01 139 0.181587 -0.946033E-01 4.06982 140 0.100056 -0.161507 141 0.176085E-01 -0.222141 8.67272 142 -0.542776E-01 -0.221212 143 0.00000 0.00000 16.5092 144 0.00000 0.00000 145 0.165459 0.170105 146 0.267422 0.925736E-01 147 0.451842 0.502135E-01 148 0.526474 0.259181E-01 149 0.559088 0.103283E-01 150 0.571536 -0.138900E-02 151 0.561680 -0.141363E-01 152 0.516587 -0.310530E-01 153 0.441663 -0.605785E-01 154 0.339303 -0.967744E-01 155 0.153200 -0.153799 156 0.00000 0.00000 157 1.00000 0.00000 -25.6472 158 1.00000 0.00000 159 1.00000 0.00000 -1.93648 160 1.00000 0.00000 161 1.00000 0.00000 0.289823 162 1.00000 0.00000 163 1.00000 0.00000 1.86864 164 1.00000 0.00000 165 1.00000 0.00000 2.99201 166 1.00000 0.00000 167 1.00000 0.00000 7.50014 168 1.00000 0.00000 169 1.00000 0.00000 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 = 0.302317E-01 l2-norm of Newton correction = 0.379819 l2-norm of FEM residual = 3.67143 l2-norm of adjusted FEM residual = 0.678595E-06 Convergence. Navier Stokes Pressures written to "navier_stokes_pressure3.txt". Navier Stokes velocities written to "navier_stokes_velocity6.txt". FREE_FEM_NAVIER_STOKES: Normal end of execution. October 7 2006 3:09:40.954 PM