To get started, select MATLAB Help or Demos from the Help menu. >> free_fem_stokes ( 'nodes6.txt', 'triangles6.txt' ) 26-Mar-2007 16:53:51 FREE_FEM_STOKES MATLAB version: Finite element solution of the steady incompressible Stokes equations on a triangulated region in 2 dimensions. - nu * ( Uxx + Uyy ) + dPdx = F1(x,y) - nu * ( Vxx + Vyy ) + 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. Quadrature order = 7 Kinematic viscosity NU = 0.010000 Node file is "nodes6.txt". Element file is "triangles6.txt". Number of nodes = 169 First 10 nodes Row: 1 2 Col 1 0.000000 0.000000 2 0.083300 0.000000 3 0.166700 0.000000 4 0.250000 0.000000 5 0.333300 0.000000 6 0.416700 0.000000 7 0.500000 0.000000 8 0.583300 0.000000 9 0.666700 0.000000 10 0.750000 0.000000 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 Dirichlet boundary condition on pressure will be applied at node 1 Number of Neumann conditions added = 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_index V_index P_index 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 Initial block of finite element matrix A: Col: 1 2 3 4 5 Row --- 1 0.010014 0.000000 0.000010 -0.006665 0.000000 2 0.000000 0.010014 0.000010 0.000000 -0.006665 3 -0.027830 -0.027830 0.000000 0.027837 -0.027810 4 -0.006665 0.000000 -0.027810 0.026648 0.000000 5 0.000000 -0.006665 -0.027810 0.000000 0.026648 6 0.001664 0.000000 -0.000007 -0.006655 0.000000 7 0.000000 0.001664 -0.000007 0.000000 -0.006655 8 -0.000027 -0.000027 0.000000 -0.027703 -0.055513 9 0.000000 0.000000 0.000000 0.000000 0.000000 10 0.000000 0.000000 0.000000 0.000000 0.000000 Col: 6 7 8 9 10 Row --- 1 0.001664 0.000000 -0.000010 0.000000 0.000000 2 0.000000 0.001664 0.000000 0.000000 0.000000 3 0.000010 0.000000 0.000000 0.000000 0.000000 4 -0.006655 0.000000 0.027810 0.000000 0.000000 5 0.000000 -0.006655 0.000000 0.000000 0.000000 6 0.020007 0.000000 0.000012 -0.006667 0.000000 7 0.000000 0.020007 0.000012 0.000000 -0.006667 8 -0.000052 -0.055618 0.000000 0.027803 -0.027787 9 -0.006667 0.000000 -0.027803 0.026667 0.000000 10 0.000000 -0.006667 -0.027787 0.000000 0.026667 Part of finite element right hand side vector: 1 0.000003 2 -0.000003 3 0.000000 4 -0.000010 5 0.000070 6 0.000032 7 0.000058 8 0.000000 9 0.000170 10 0.000124 Matrix A after Dirichlet BC adjustments: Col: 1 2 3 4 5 Row --- 1 1.000000 0.000000 0.000000 0.000000 0.000000 2 0.000000 1.000000 0.000000 0.000000 0.000000 3 0.000000 0.000000 1.000000 0.000000 0.000000 4 0.000000 0.000000 0.000000 1.000000 0.000000 5 0.000000 0.000000 0.000000 0.000000 1.000000 6 0.000000 0.000000 0.000000 0.000000 0.000000 7 0.000000 0.000000 0.000000 0.000000 0.000000 8 -0.000027 -0.000027 0.000000 -0.027703 -0.055513 9 0.000000 0.000000 0.000000 0.000000 0.000000 10 0.000000 0.000000 0.000000 0.000000 0.000000 Col: 6 7 8 9 10 Row --- 1 0.000000 0.000000 0.000000 0.000000 0.000000 2 0.000000 0.000000 0.000000 0.000000 0.000000 3 0.000000 0.000000 0.000000 0.000000 0.000000 4 0.000000 0.000000 0.000000 0.000000 0.000000 5 0.000000 0.000000 0.000000 0.000000 0.000000 6 1.000000 0.000000 0.000000 0.000000 0.000000 7 0.000000 1.000000 0.000000 0.000000 0.000000 8 -0.000052 -0.055618 0.000000 0.027803 -0.027787 9 0.000000 0.000000 0.000000 1.000000 0.000000 10 0.000000 0.000000 0.000000 0.000000 1.000000 Part of right hand side after Dirichlet BC adjustment: 1 0.000000 2 0.000000 3 0.000000 4 0.000000 5 0.000000 6 0.000000 7 0.000000 8 0.000000 9 0.000000 10 0.000000 Part of the solution vector: 1 0.000000 2 0.000000 3 0.000000 4 0.000000 5 0.000000 6 0.000000 7 0.000000 8 0.000456 9 0.000000 10 0.000000 Variable indices per node: Node U V P 1 0.000000 0.000000 0.000000 2 0.000000 0.000000 3 0.000000 0.000000 0.000456 4 0.000000 0.000000 5 0.000000 0.000000 0.000029 6 0.000000 0.000000 7 0.000000 0.000000 -0.000044 8 0.000000 0.000000 9 0.000000 0.000000 -0.000143 10 0.000000 0.000000 11 0.000000 0.000000 -0.000133 12 0.000000 0.000000 13 0.000000 0.000000 0.000162 14 0.000000 0.000000 15 0.006754 -0.006754 16 0.023469 -0.009712 17 0.042630 -0.009714 18 0.058270 -0.007246 19 0.067416 -0.003182 20 0.068626 0.001343 21 0.061728 0.005308 22 0.048757 0.008028 23 0.032541 0.009045 24 0.016496 0.008040 25 0.004599 0.005060 26 0.000000 0.000000 27 0.000000 0.000000 -0.000456 28 0.009712 -0.023469 29 0.033091 -0.033091 0.000000 30 0.060392 -0.032297 31 0.083455 -0.022715 0.000016 32 0.096025 -0.009559 33 0.098282 0.005751 0.000014 34 0.087764 0.018261 35 0.069641 0.027183 0.000017 36 0.046109 0.029918 37 0.023035 0.025898 -0.000025 38 0.006466 0.016034 39 0.000000 0.000000 0.000069 40 0.000000 0.000000 41 0.009714 -0.042630 42 0.032297 -0.060392 43 0.058168 -0.058168 44 0.079813 -0.041625 45 0.092335 -0.016969 46 0.093935 0.009843 47 0.084319 0.033125 48 0.066609 0.048874 49 0.044300 0.053849 50 0.022393 0.046957 51 0.006194 0.028436 52 0.000000 0.000000 53 0.000000 0.000000 -0.000029 54 0.007246 -0.058270 55 0.022715 -0.083455 -0.000016 56 0.041625 -0.079813 57 0.057045 -0.057045 0.000000 58 0.066079 -0.023078 59 0.067443 0.013961 -0.000011 60 0.060302 0.045612 61 0.047965 0.067555 -0.000004 62 0.031618 0.073920 63 0.016142 0.064665 -0.000016 64 0.004358 0.038862 65 0.000000 0.000000 0.000034 66 0.000000 0.000000 67 0.003182 -0.067416 68 0.009559 -0.096025 69 0.016969 -0.092335 70 0.023078 -0.066079 71 0.026751 -0.026751 72 0.027095 0.015769 73 0.024267 0.052778 74 0.019140 0.077722 75 0.012640 0.085509 76 0.006352 0.074521 77 0.001666 0.044904 78 0.000000 0.000000 79 0.000000 0.000000 0.000044 80 -0.001343 -0.068626 81 -0.005751 -0.098282 -0.000014 82 -0.009843 -0.093935 83 -0.013961 -0.067443 0.000011 84 -0.015769 -0.027095 85 -0.016148 0.016148 -0.000000 86 -0.014535 0.053769 87 -0.011274 0.079383 -0.000002 88 -0.007717 0.087034 89 -0.003616 0.076184 -0.000005 90 -0.001173 0.045602 91 0.000000 0.000000 -0.000027 92 0.000000 0.000000 93 -0.005308 -0.061728 94 -0.018261 -0.087764 95 -0.033125 -0.084319 96 -0.045612 -0.060302 97 -0.052778 -0.024267 98 -0.053769 0.014535 99 -0.048396 0.048396 100 -0.038214 0.071107 101 -0.025548 0.078196 102 -0.012985 0.068084 103 -0.003690 0.040923 104 0.000000 0.000000 105 0.000000 0.000000 0.000143 106 -0.008028 -0.048757 107 -0.027183 -0.069641 -0.000017 108 -0.048874 -0.066609 109 -0.067555 -0.047965 0.000004 110 -0.077722 -0.019140 111 -0.079383 0.011274 0.000002 112 -0.071107 0.038214 113 -0.056180 0.056180 0.000000 114 -0.037402 0.061765 115 -0.018700 0.054034 0.000009 116 -0.005297 0.032282 117 0.000000 0.000000 -0.000044 118 0.000000 0.000000 119 -0.009045 -0.032541 120 -0.029918 -0.046109 121 -0.053849 -0.044300 122 -0.073920 -0.031618 123 -0.085509 -0.012640 124 -0.087034 0.007717 125 -0.078196 0.025548 126 -0.061765 0.037402 127 -0.041150 0.041150 128 -0.020862 0.035759 129 -0.005789 0.021503 130 0.000000 0.000000 131 0.000000 0.000000 0.000133 132 -0.008040 -0.016496 133 -0.025898 -0.023035 0.000025 134 -0.046957 -0.022393 135 -0.064665 -0.016142 0.000016 136 -0.074521 -0.006352 137 -0.076184 0.003616 0.000005 138 -0.068084 0.012985 139 -0.054034 0.018700 -0.000009 140 -0.035759 0.020862 141 -0.018025 0.018025 0.000000 142 -0.004937 0.010819 143 0.000000 0.000000 -0.000097 144 0.000000 0.000000 145 -0.005060 -0.004599 146 -0.016034 -0.006466 147 -0.028436 -0.006194 148 -0.038862 -0.004358 149 -0.044904 -0.001666 150 -0.045602 0.001173 151 -0.040923 0.003690 152 -0.032282 0.005297 153 -0.021503 0.005789 154 -0.010819 0.004937 155 -0.002924 0.002924 156 0.000000 0.000000 157 0.000000 0.000000 -0.000162 158 0.000000 0.000000 159 0.000000 0.000000 -0.000069 160 0.000000 0.000000 161 0.000000 0.000000 -0.000034 162 0.000000 0.000000 163 0.000000 0.000000 0.000027 164 0.000000 0.000000 165 0.000000 0.000000 0.000044 166 0.000000 0.000000 167 0.000000 0.000000 0.000097 168 0.000000 0.000000 169 0.000000 0.000000 0.000000 Pressure nodes written to "nodes3.txt". Pressure triangles written to "triangles3.txt". Pressures written to "pressure3.txt". FREE_FEM_STOKES: Normal end of execution. 26-Mar-2007 16:54:01 >>