def darcy_dual_hybridVEM_example3(**kwargs): ####################### # Simple 3d Darcy problem with known exact solution ####################### Nx = Ny = Nz = 7 g = structured.CartGrid([Nx, Ny, Nz], [1, 1, 1]) g.compute_geometry() kxx = np.ones(g.num_cells) perm = tensor.SecondOrder(g.dim, kxx) def funP_ex(pt): return np.sin(2*np.pi*pt[0])*np.sin(2*np.pi*pt[1])\ * np.sin(2*np.pi*pt[2]) def funU_ex(pt): return [-2*np.pi*np.cos(2*np.pi*pt[0])\ * np.sin(2*np.pi*pt[1])*np.sin(2*np.pi*pt[2]), -2*np.pi*np.sin(2*np.pi*pt[0])\ * np.cos(2*np.pi*pt[1])*np.sin(2*np.pi*pt[2]), -2*np.pi*np.sin(2*np.pi*pt[0])\ * np.sin(2*np.pi*pt[1])*np.cos(2*np.pi*pt[2])] def fun(pt): return 12 * np.pi**2 * funP_ex(pt) f = np.array([fun(pt) for pt in g.cell_centers.T]) b_faces = g.get_boundary_faces() bnd = bc.BoundaryCondition(g, b_faces, ['dir'] * b_faces.size) bnd_val = np.zeros(g.num_faces) bnd_val[b_faces] = funP_ex(g.face_centers[:, b_faces]) solver = hybrid.HybridDualVEM() data = {'perm': perm, 'source': f, 'bc': bnd, 'bc_val': bnd_val} H, rhs = solver.matrix_rhs(g, data) l = sps.linalg.spsolve(H, rhs) u, p = solver.compute_up(g, l, data) P0u = dual.DualVEM().project_u(g, u) if kwargs['visualize']: plot_grid(g, p, P0u) p_ex = error.interpolate(g, funP_ex) u_ex = error.interpolate(g, funU_ex) np.set_printoptions(linewidth=999999) np.set_printoptions(precision=16) errors = np.array( [error.error_L2(g, p, p_ex), error.error_L2(g, P0u, u_ex)]) errors_known = np.array([0.1010936831876412, 0.0680593765009036]) assert np.allclose(errors, errors_known)
def darcy_dual_hybridVEM_example2(**kwargs): ####################### # Simple 2d Darcy problem on a surface with known exact solution ####################### Nx = Ny = 25 g = simplex.StructuredTriangleGrid([Nx, Ny], [1, 1]) R = cg.rot(np.pi / 6., [0, 1, 1]) g.nodes = np.dot(R, g.nodes) g.compute_geometry(is_embedded=True) T = cg.tangent_matrix(g.nodes) kxx = np.ones(g.num_cells) perm = tensor.SecondOrder(g.dim, kxx) def funP_ex(pt): return np.pi * pt[0] - 6 * pt[1] + np.exp(1) * pt[2] - 4 def funU_ex(pt): return np.dot(T, [-np.pi, 6, -np.exp(1)]) def fun(pt): return 0 f = np.array([fun(pt) for pt in g.cell_centers.T]) b_faces = g.get_boundary_faces() bnd = bc.BoundaryCondition(g, b_faces, ['dir'] * b_faces.size) bnd_val = np.zeros(g.num_faces) bnd_val[b_faces] = funP_ex(g.face_centers[:, b_faces]) solver = hybrid.HybridDualVEM() data = {'perm': perm, 'source': f, 'bc': bnd, 'bc_val': bnd_val} H, rhs = solver.matrix_rhs(g, data) l = sps.linalg.spsolve(H, rhs) u, p = solver.compute_up(g, l, data) P0u = dual.DualVEM().project_u(g, u) if kwargs['visualize']: plot_grid(g, p, P0u) p_ex = error.interpolate(g, funP_ex) u_ex = error.interpolate(g, funU_ex) errors = np.array( [error.error_L2(g, p, p_ex), error.error_L2(g, P0u, u_ex)]) errors_known = np.array([0, 0]) assert np.allclose(errors, errors_known)
def darcy_dualVEM_example1(**kwargs): ####################### # Simple 2d Darcy problem with known exact solution ####################### Nx = Ny = 25 g = structured.CartGrid([Nx, Ny], [1, 1]) g.compute_geometry() kxx = np.ones(g.num_cells) perm = tensor.SecondOrder(g.dim, kxx) def funP_ex(pt): return np.sin(2 * np.pi * pt[0]) * np.sin(2 * np.pi * pt[1]) def funU_ex(pt): return [ -2 * np.pi * np.cos(2 * np.pi * pt[0]) * np.sin(2 * np.pi * pt[1]), -2 * np.pi * np.sin(2 * np.pi * pt[0]) * np.cos(2 * np.pi * pt[1]), 0 ] def fun(pt): return 8 * np.pi**2 * funP_ex(pt) f = np.array([fun(pt) for pt in g.cell_centers.T]) b_faces = g.get_boundary_faces() bnd = bc.BoundaryCondition(g, b_faces, ['dir'] * b_faces.size) bnd_val = np.zeros(g.num_faces) bnd_val[b_faces] = funP_ex(g.face_centers[:, b_faces]) solver = dual.DualVEM() data = {'perm': perm, 'source': f, 'bc': bnd, 'bc_val': bnd_val} D, rhs = solver.matrix_rhs(g, data) up = sps.linalg.spsolve(D, rhs) u, p = solver.extract_u(g, up), solver.extract_p(g, up) P0u = solver.project_u(g, u) if kwargs['visualize']: plot_grid(g, p, P0u) p_ex = error.interpolate(g, funP_ex) u_ex = error.interpolate(g, funU_ex) errors = np.array( [error.error_L2(g, p, p_ex), error.error_L2(g, P0u, u_ex)]) errors_known = np.array([0.0210718223032, 0.00526933885613]) assert np.allclose(errors, errors_known)