## [501e5e]: devel / example / Ficticious_Domain / Steady_state / H1_penalization / NS_with_H1_penalization.m Maximize Restore History

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151``` ```## Copyright (C) 2013 Marco Vassallo ## ## This program is free software; you can redistribute it and/or modify it under ## the terms of the GNU General Public License as published by the Free Software ## Foundation; either version 3 of the License, or (at your option) any later ## version. ## ## This program is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more ## details. ## ## You should have received a copy of the GNU General Public License along with ## this program; if not, see . # In this example we solve the steady 2D NS equation for a flow around a square # cylinder in a channel. In this example, the no-slip BC are applied using # a H1 penalization technique. # We use the preconditioned gmres algorithm. pkg load fem-fenics msh x = linspace (0, 4, 33); y = linspace (0, 1, 9); msho = msh2m_structured_mesh (x, y, 1, 1:4); mesh = Mesh (msho); plot (mesh); import_ufl_Problem ('A_H1'); import_ufl_BilinearForm ('B_H1'); import_ufl_BilinearForm ('C_H1'); import_ufl_Functional ('Err_u'); import_ufl_Functional ('Err_p'); import_ufl_FunctionSpace ('C_H1'); TH1 = FunctionSpace ('A_H1', mesh); TH2 = FunctionSpace ('C_H1', mesh); bc1 = DirichletBC (TH1, @(x, y, z, n) [0, 0], [1, 3]); bc2 = DirichletBC (TH1, @(x, y) [4*(1 - y)*(y) , 0], 4); bc = {bc1, bc2}; u0 = Expression ('u0', @(x, y) [0; 0]); f = Expression ('f', @(x, y) [0; 0]); nu_1 = 1/40; nu_0 = 1/40; r = 0.25; #ficticious domain dom = @(x, y) (x <= 1+r)*(x >= 1)*(y >= (0.5 - r/2))*(y <= (0.5 + r/2)); nu = Expression ('nu', @(x, y) dom(x, y) * nu_0 + ... (1 - dom(x, y)) * nu_1); sig_1 = 0; sig_0 = 1e4; sig = Expression ('sig',@(x, y) dom(x, y) * sig_0 + ... (1 - dom(x, y)) * sig_1); a = BilinearForm ('A_H1', TH1, TH1, nu, sig, u0); L = LinearForm ('A_H1', TH1, f); [A, ff] = assemble_system (a, L, bc{:}); b = BilinearForm ('B_H1', TH1, TH2); B = assemble(b); m = BilinearForm ('C_H1', TH2, TH2, nu); M = assemble(m); [x1, y1, v1] = find (A); [x2, y2, v2] = find (B'); y2 += size (A, 1); [x3, y3, v3] = find (B); x3 += size (A, 1); [x4, y4, v4] = find (M); x4 += size (A, 1); y4 += size (A, 1); C = sparse ([x1; x2; x3],[y1; y2; y3],[v1; v2; v3], (size (A,1) + size (B,1)), (size (A, 1) + size (B, 1))); P = sparse ([x1; x4],[y1; y4],[v1; v4], (size (A,1) + size (B,1)), (size (A, 1) + size (B, 1))); F = [ff; (zeros (size (B, 1), 1))]; [sol, flag, relres, iter, resvec] = gmres (C, F, [], 1e-6, 100, P); fprintf('Gmres converges in %d Iteration\n',iter (2)); u = Function ('u', TH1, sol(1: (size(A,1)))); p = Function ('p', TH2, sol((size(A,1))+1 : end)); save (u, 'velocity'); save (p, 'pressure'); #Compute the initial norm p0 = Expression ('p0', @(x, y) 0); E1 = Functional ('Err_u', mesh, u, u0); normu0 = sqrt (assemble(E1)); E2 = Functional ('Err_p', mesh, p, p0); normp0 = sqrt (assemble(E2)); u0 = Function('u0',TH1, sol(1: (size(A,1)))); #Iteration err = 10; tol = 1e-4; maxit = 100; i = 1; while (err > tol && i < maxit) a = BilinearForm ('A_H1', TH1, TH1, nu, sig, u0); [A, ff] = assemble_system (a, L, bc{:}); [x1, y1, v1] = find (A); C = sparse ([x1; x2; x3],[y1; y2; y3],[v1; v2; v3], (size (A,1) + size (B,1)), (size (A, 1) + size (B, 1))); F = [ff; (zeros (size (B, 1), 1))]; [sol, flag, relres, iter, resvec] = gmres (C, F, 100, 1e-6, 2000, P); fprintf('iteration %d: Gmres converges in %d Iteration\n',i, iter (2)); u = Function ('u', TH1, sol(1: (size(A,1)))); p = Function ('p', TH2, sol((size(A,1))+1 : end)); E1 = Functional ('Err_u', mesh, u, u0); normu = sqrt (assemble(E1)); E2 = Functional ('Err_p', mesh, p, p0); normp = sqrt (assemble(E2)); err = normu/normu0 + normp/normp0; fprintf('the error is %f \n',err); pause (0); i++; u0 = Function ('u0',TH1, sol(1: (size(A,1)))); p0 = Function ('p0', TH2, sol((size(A,1))+1 : end)); save (u, 'velocity'); save (p, 'pressure'); end plot (u); norm_err = 0; for i = 1:size(msho.p, 2) x = msho.p (1, i); y = msho.p (2, i); if (dom (x, y) == true) err_L2 = feval (u, [x, y]); norm_err += err_L2(1).^2 + err_L2(2).^2; endif endfor error_on_bc = sqrt (norm_err) ```