>> ffns_sparse ( 'nodes6.txt', 'triangles6.txt' ) 24-Sep-2006 12:23:16 FFNS_SPARSE (MATLAB version): This is a special version of FREE_FEM_NAVIER_STOKES which uses MATLAB sparse matrix storage, factorization, and solution methods, replacing the general banded approach. 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: Fluid viscosity NU = 1.000000: Node file is "nodes6.txt". Element file is "triangles6.txt". Number of nodes = 65 First 10 nodes Row: 1 2 Col 1 0.000000 0.000000 2 0.000000 0.250000 3 0.000000 0.500000 4 0.000000 0.750000 5 0.000000 1.000000 6 0.250000 0.000000 7 0.250000 0.250000 8 0.250000 0.500000 9 0.250000 0.750000 10 0.250000 1.000000 Element order = 6 Number of elements = 24 First 10 elements Row: 1 2 3 4 5 6 Col 1 1 11 3 6 7 2 2 13 3 11 8 7 12 3 3 13 5 8 9 4 4 15 5 13 10 9 14 5 11 21 13 16 17 12 6 23 13 21 18 17 22 7 13 23 15 18 19 14 8 25 15 23 20 19 24 9 21 31 23 26 27 22 10 33 23 31 28 27 32 Quadrature order = 3 Dirichlet boundary condition on pressure will be applied at node 1 Total number of variables is 151 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 -1 8 18 19 -1 9 20 21 -1 10 22 23 -1 11 24 25 26 12 27 28 -1 13 29 30 31 14 32 33 -1 15 34 35 36 16 37 38 -1 17 39 40 -1 18 41 42 -1 19 43 44 -1 20 45 46 -1 21 47 48 49 22 50 51 -1 23 52 53 54 24 55 56 -1 25 57 58 59 26 60 61 -1 27 62 63 -1 28 64 65 -1 29 66 67 -1 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 -1 38 87 88 -1 39 89 90 -1 40 91 92 -1 41 93 94 95 42 96 97 -1 43 98 99 100 44 101 102 -1 45 103 104 105 46 106 107 -1 47 108 109 -1 48 110 111 -1 49 112 113 -1 50 114 115 -1 51 116 117 118 52 119 120 -1 53 121 122 123 54 124 125 -1 55 126 127 128 56 129 130 -1 57 131 132 -1 58 133 134 -1 59 135 136 -1 60 137 138 -1 61 139 140 141 62 142 143 -1 63 144 145 146 64 147 148 -1 65 149 150 151 Order 6 nodes plotted in "nodes6.eps". Order 6 triangles plotted in "triangles6.eps". Initial block of Stokes stiffness matrix A: Col: 1 2 3 4 5 Row 1 1 0 0 -0.666667 0 2 0 1 0 0 -0.666667 3 -0.083333 -0.083333 0 -0.083333 0.083333 4 -0.666667 0 -0.083333 2.666667 0 5 0 -0.666667 -0.083333 0 2.666667 6 0.166667 0 0 -0.666667 0 7 0 0.166667 0 0 -0.666667 8 0 0 0 -0.166667 -0.083333 9 0 0 0 0 0 10 0 0 0 0 0 Col: 6 7 8 9 10 Row 1 0.166667 0 0 0 0 2 0 0.166667 0 0 0 3 0 0 0 0 0 4 -0.666667 0 0 0 0 5 0 -0.666667 0.083333 0 0 6 2.000000 0 0 -0.666667 0 7 0 2.000000 0 0 -0.666667 8 -0.166667 0 0 -0.083333 0.083333 9 -0.666667 0 -0.083333 2.666667 0 10 0 -0.666667 -0.083333 0 2.666667 Part of Stokes right hand side vector: 1 0.000000 2 0.000000 3 0.000000 4 0.000000 5 0.000000 6 0.000000 7 0.000000 8 0.000000 ...... .............. 151 0.000000 Stokes matrix after boundary condition adjustments: Col: 1 2 3 4 5 Row 1 1 0 0 0 0 2 0 1 0 0 0 3 0 0 1 0 0 4 0 0 0 1 0 5 0 0 0 0 1 6 0 0 0 0 0 7 0 0 0 0 0 8 0 0 0 -0.166667 -0.083333 9 0 0 0 0 0 10 0 0 0 0 0 Col: 6 7 8 9 10 Row 1 0 0 0 0 0 2 0 0 0 0 0 3 0 0 0 0 0 4 0 0 0 0 0 5 0 0 0 0 0 6 1 0 0 0 0 7 0 1 0 0 0 8 -0.166667 0 0 -0.083333 0.083333 9 0 0 0 1 0 10 0 0 0 0 1 Part of adjusted Stokes right hand: 1 0.000000 2 0.000000 3 0.000000 4 0.187500 5 0.000000 6 0.250000 7 0.000000 8 0.000000 ...... .............. 151 0.000000 Part of the solution vector: 1 0.000000 2 0.000000 3 0.000000 4 0.187500 5 0.000000 6 0.250000 7 0.000000 8 -0.000000 ...... .............. 151 -6.000000 Solution to the STOKES equations: Node U V P 1 0.000000 0.000000 0.000000 2 0.187500 0.000000 3 0.250000 0.000000 -0.000000 4 0.187500 0.000000 5 0.000000 0.000000 0.000000 6 0.000000 0.000000 7 0.187500 0.000000 8 0.250000 -0.000000 9 0.187500 -0.000000 10 0.000000 0.000000 11 0.000000 0.000000 -1.000000 12 0.187500 0.000000 13 0.250000 0.000000 -1.000000 14 0.187500 0.000000 15 0.000000 0.000000 -1.000000 16 0.000000 0.000000 17 0.187500 -0.000000 18 0.250000 0.000000 19 0.187500 0.000000 20 0.000000 0.000000 21 0.000000 0.000000 -2.000000 22 0.187500 -0.000000 23 0.250000 -0.000000 -2.000000 24 0.187500 -0.000000 25 0.000000 0.000000 -2.000000 26 0.000000 0.000000 27 0.187500 -0.000000 28 0.250000 -0.000000 29 0.187500 -0.000000 30 0.000000 0.000000 31 0.000000 0.000000 -3.000000 32 0.187500 0.000000 33 0.250000 -0.000000 -3.000000 34 0.187500 0.000000 35 0.000000 0.000000 -3.000000 36 0.000000 0.000000 37 0.187500 0.000000 38 0.250000 -0.000000 39 0.187500 -0.000000 40 0.000000 0.000000 41 0.000000 0.000000 -4.000000 42 0.187500 0.000000 43 0.250000 0.000000 -4.000000 44 0.187500 -0.000000 45 0.000000 0.000000 -4.000000 46 0.000000 0.000000 47 0.187500 0.000000 48 0.250000 0.000000 49 0.187500 0.000000 50 0.000000 0.000000 51 0.000000 0.000000 -5.000000 52 0.187500 -0.000000 53 0.250000 -0.000000 -5.000000 54 0.187500 0.000000 55 0.000000 0.000000 -5.000000 56 0.000000 0.000000 57 0.187500 -0.000000 58 0.250000 -0.000000 59 0.187500 0.000000 60 0.000000 0.000000 61 0.000000 0.000000 -6.000000 62 0.187500 0.000000 63 0.250000 0.000000 -6.000000 64 0.187500 0.000000 65 0.000000 0.000000 -6.000000 Pressure nodes written to "nodes3.txt". Pressure triangles written to "triangles3.txt". Pressures written to "stokes_pressure3.txt". Part of Navier-Stokes FEM residual: 1 -0.083333 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 11 -0.083333 12 0.000000 13 0.000000 14 -0.333333 15 -0.000000 16 0.000000 17 0.000000 18 0.000000 19 0.000000 20 0.000000 21 -0.000000 22 -0.333333 23 -0.000000 24 -0.166667 25 0.000000 26 0.000000 27 -0.000000 28 -0.000000 29 0.000000 30 0.000000 31 0.000000 32 -0.000000 33 -0.000000 34 -0.166667 35 -0.000000 36 -0.000000 37 -0.333333 38 0.000000 39 0.000000 40 0.000000 41 -0.000000 42 0.000000 43 0.000000 44 0.000000 45 -0.333333 46 0.000000 47 -0.166667 48 0.000000 49 -0.000000 50 0.000000 51 -0.000000 52 0.000000 53 -0.000000 54 -0.000000 55 -0.000000 56 0.000000 57 -0.166667 58 0.000000 59 0.000000 60 -0.333333 61 0.000000 62 0.000000 63 0.000000 64 0.000000 65 -0.000000 66 0.000000 67 -0.000000 68 -0.333333 69 -0.000000 70 -0.166667 71 -0.000000 72 0.000000 73 0.000000 74 0.000000 75 -0.000000 76 0.000000 77 -0.000000 78 0.000000 79 -0.000000 80 -0.166667 81 -0.000000 82 -0.000000 83 -0.333333 84 -0.000000 85 0.000000 86 -0.000000 87 0.000000 88 -0.000000 89 -0.000000 90 -0.000000 91 -0.333333 92 0.000000 93 -0.166667 94 -0.000000 95 -0.000000 96 -0.000000 97 -0.000000 98 -0.000000 99 -0.000000 100 0.000000 101 0.000000 102 -0.000000 103 -0.166667 104 0.000000 105 -0.000000 106 -0.333333 107 -0.000000 108 0.000000 109 -0.000000 110 -0.000000 111 -0.000000 112 0.000000 113 0.000000 114 -0.333333 115 0.000000 116 -0.166667 117 0.000000 118 0.000000 119 -0.000000 120 -0.000000 121 -0.000000 122 -0.000000 123 0.000000 124 -0.000000 125 0.000000 126 -0.166667 127 -0.000000 128 0.000000 129 -0.333333 130 -0.000000 131 0.000000 132 -0.000000 133 0.000000 134 0.000000 135 0.000000 136 0.000000 137 -0.333333 138 -0.000000 139 -0.083333 140 0.000000 141 -0.000000 142 0.000000 143 0.000000 144 0.000000 145 0.000000 146 0.000000 147 -0.000000 148 -0.000000 149 -0.083333 150 0.000000 151 -0.000000 l2-norm of FEM residual = 0.824958 Part of Navier-Stokes FEM residual adjusted for BC: 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 11 0.000000 12 0.000000 13 0.000000 14 0.000000 15 0.000000 16 0.000000 17 0.000000 18 0.000000 19 0.000000 20 0.000000 21 -0.000000 22 0.000000 23 0.000000 24 0.000000 25 0.000000 26 0.000000 27 -0.000000 28 -0.000000 29 0.000000 30 0.000000 31 0.000000 32 -0.000000 33 -0.000000 34 0.000000 35 0.000000 36 -0.000000 37 0.000000 38 0.000000 39 0.000000 40 0.000000 41 -0.000000 42 0.000000 43 0.000000 44 0.000000 45 0.000000 46 0.000000 47 0.000000 48 0.000000 49 -0.000000 50 0.000000 51 -0.000000 52 0.000000 53 -0.000000 54 -0.000000 55 -0.000000 56 0.000000 57 0.000000 58 0.000000 59 0.000000 60 0.000000 61 0.000000 62 0.000000 63 0.000000 64 0.000000 65 -0.000000 66 0.000000 67 -0.000000 68 0.000000 69 0.000000 70 0.000000 71 0.000000 72 0.000000 73 0.000000 74 0.000000 75 -0.000000 76 0.000000 77 -0.000000 78 0.000000 79 -0.000000 80 0.000000 81 0.000000 82 -0.000000 83 0.000000 84 0.000000 85 0.000000 86 -0.000000 87 0.000000 88 -0.000000 89 -0.000000 90 -0.000000 91 0.000000 92 0.000000 93 0.000000 94 0.000000 95 -0.000000 96 -0.000000 97 -0.000000 98 -0.000000 99 -0.000000 100 0.000000 101 0.000000 102 -0.000000 103 0.000000 104 0.000000 105 -0.000000 106 0.000000 107 0.000000 108 0.000000 109 -0.000000 110 -0.000000 111 -0.000000 112 0.000000 113 0.000000 114 0.000000 115 0.000000 116 0.000000 117 0.000000 118 0.000000 119 -0.000000 120 -0.000000 121 -0.000000 122 -0.000000 123 0.000000 124 -0.000000 125 0.000000 126 0.000000 127 0.000000 128 0.000000 129 0.000000 130 0.000000 131 0.000000 132 -0.000000 133 0.000000 134 0.000000 135 0.000000 136 0.000000 137 0.000000 138 0.000000 139 0.000000 140 0.000000 141 -0.000000 142 0.000000 143 0.000000 144 0.000000 145 0.000000 146 0.000000 147 0.000000 148 0.000000 149 0.000000 150 0.000000 151 -0.000000 l2-norm of adjusted FEM residual = 0.000000 Convergence. Navier Stokes Pressures written to "navier_stokes_pressure3.txt". Navier Stokes velocities written to "navier_stokes_velocity6.txt". FFNS_SPARSE: Normal end of execution. 24-Sep-2006 12:23:27 >>