def test_multiple_elements(self): """Test the anysotropic case in multiple element.""" p = (6, 6) is_inner = False crazy_mesh = CrazyMesh(2, (8, 5), ((-1, 1), (-1, 1)), curvature=0.1) func_space_2_lobatto = FunctionSpace(crazy_mesh, '2-lobatto', p, is_inner) func_space_1_lobatto = FunctionSpace(crazy_mesh, '1-lobatto', p, is_inner) func_space_1_lobatto.dof_map.continous_dof = True def diffusion_11(x, y): return 4 * np.ones(np.shape(x)) def diffusion_12(x, y): return 3 * np.ones(np.shape(x)) def diffusion_22(x, y): return 5 * np.ones(np.shape(x)) def source(x, y): return -36 * np.pi ** 2 * np.sin(2 * np.pi * x) * np.sin(2 * np.pi * y) + 24 * np.pi ** 2 * np.cos(2 * np.pi * x) * np.cos(2 * np.pi * y) # mesh function to inject the anisotropic tensor mesh_k = MeshFunction(crazy_mesh) mesh_k.continous_tensor = [diffusion_11, diffusion_12, diffusion_22] # definition of the basis functions basis_1 = BasisForm(func_space_1_lobatto) basis_1.quad_grid = 'gauss' basis_2 = BasisForm(func_space_2_lobatto) basis_2.quad_grid = 'gauss' # solution form phi_2 = Form(func_space_2_lobatto) phi_2.basis.quad_grid = 'gauss' form_source = Form(func_space_2_lobatto) form_source.discretize(source, ('gauss', 30)) # find inner product M_1k = inner(basis_1, basis_1, mesh_k) N_2 = inner(d(basis_1), basis_2) M_2 = inner(basis_2, basis_2) # assemble M_1k = assemble(M_1k, func_space_1_lobatto) N_2 = assemble(N_2, (func_space_1_lobatto, func_space_2_lobatto)) M_2 = assemble(M_2, func_space_2_lobatto) lhs = sparse.bmat([[M_1k, N_2], [N_2.transpose(), None]]).tocsc() rhs_source = (form_source.cochain @ M_2)[:, np.newaxis] rhs_zeros = np.zeros(lhs.shape[0] - np.size(rhs_source))[:, np.newaxis] rhs = np.vstack((rhs_zeros, rhs_source)) solution = sparse.linalg.spsolve(lhs, rhs) phi_2.cochain = solution[-func_space_2_lobatto.num_dof:] # sample the solution xi = eta = np.linspace(-1, 1, 200) phi_2.reconstruct(xi, eta) (x, y), data = phi_2.export_to_plot() plt.contourf(x, y, data) plt.show() print("max value {0} \nmin value {1}" .format(np.max(data), np.min(data))) npt.assert_array_almost_equal(self.solution(x, y), data, decimal=2)
def test_single_element(self): """Test the anysotropic case in a single element.""" dim = 2 elements_layout = (1, 1) bounds_domain = ((-1, 1), (-1, 1)) curvature = 0.1 p = (20, 20) is_inner = False crazy_mesh = CrazyMesh(dim, elements_layout, bounds_domain, curvature) func_space_2_lobatto = FunctionSpace(crazy_mesh, '2-lobatto', p, is_inner) func_space_1_lobatto = FunctionSpace(crazy_mesh, '1-lobatto', p, is_inner) def diffusion_11(x, y): return 4 * np.ones(np.shape(x)) def diffusion_12(x, y): return 3 * np.ones(np.shape(x)) def diffusion_22(x, y): return 5 * np.ones(np.shape(x)) mesh_k = MeshFunction(crazy_mesh) mesh_k.continous_tensor = [diffusion_11, diffusion_12, diffusion_22] basis_1 = BasisForm(func_space_1_lobatto) basis_1.quad_grid = 'gauss' basis_2 = BasisForm(func_space_2_lobatto) basis_2.quad_grid = 'gauss' phi_2 = Form(func_space_2_lobatto) phi_2.basis.quad_grid = 'gauss' M_1k = inner(basis_1, basis_1, mesh_k) N_2 = inner(basis_2, d(basis_1)) def source(x, y): return -36 * np.pi ** 2 * np.sin(2 * np.pi * x) * np.sin(2 * np.pi * y) + 24 * np.pi ** 2 * np.cos(2 * np.pi * x) * np.cos(2 * np.pi * y) form_source = Form(func_space_2_lobatto) form_source.discretize(source) M_2 = inner(basis_2, basis_2) lhs = np.vstack((np.hstack((M_1k[:, :, 0], N_2[:, :, 0])), np.hstack( (np.transpose(N_2[:, :, 0]), np.zeros((np.shape(N_2)[1], np.shape(N_2)[1])))))) rhs = np.zeros((np.shape(lhs)[0], 1)) rhs[-np.shape(M_2)[0]:] = form_source.cochain @ M_2 phi_2.cochain = np.linalg.solve(lhs, rhs)[-phi_2.basis.num_basis:].flatten() xi = eta = np.linspace(-1, 1, 200) phi_2.reconstruct(xi, eta) (x, y), data = phi_2.export_to_plot() print("max value {0} \nmin value {1}" .format(np.max(data), np.min(data))) npt.assert_array_almost_equal(self.solution(x, y), data, decimal=2)
def test_weighted_inner_continous(self): """Test for weighted inner product.""" mesh = CrazyMesh(2, (2, 2), ((-1, 1), (-1, 1)), curvature=0.2) func_space = FunctionSpace(mesh, '1-lobatto', (3, 4)) basis = BasisForm(func_space) basis.quad_grid = 'gauss' K = MeshFunction(mesh) K.continous_tensor = [diff_tens_11, diff_tens_12, diff_tens_22] M_1_weighted = inner(basis, basis, K) M_1 = inner(basis, basis) npt.assert_array_almost_equal(M_1, M_1_weighted)
def test_weighted_metric(self): # TODO: figure out why if the metric tensor is set to ones the result is very different """Compare weighted and unweighted metric terms with K set to identity.""" mesh = CrazyMesh(2, (1, 1), ((-1, 1), (-1, 1)), curvature=0.2) K = MeshFunction(mesh) func_space = FunctionSpace(mesh, '1-lobatto', (3, 3)) basis = BasisForm(func_space) K.continous_tensor = [diff_tens_11, diff_tens_12, diff_tens_22] xi = eta = np.linspace(-1, 1, 5) xi, eta = np.meshgrid(xi, eta) g_11_k, g_12_k, g_22_k = basis.weighted_metric_tensor(xi.ravel('F'), eta.ravel('F'), K) g_11, g_12, g_22 = mesh.metric_tensor(xi.ravel('F'), eta.ravel('F')) npt.assert_array_almost_equal(g_11, g_11_k)
def test_inner(self): """Test inner product of one forms.""" list_cases = ['p2_n2-2', 'p2_n3-2', 'p5_n1-10', 'p10_n2-2', 'p13_n12-8'] p = [2, 2, 5, 10, 13] n = [(2, 2), (3, 2), (1, 10), (2, 2), (12, 8)] curvature = [0.1, 0.1, 0.1, 0.1, 0.1] for i, case in enumerate(list_cases[:-1]): M_1_ref = np.loadtxt( os.getcwd() + '/src/tests/test_M_1/M_1k_' + case + '.dat', delimiter=',').reshape(2 * p[i] * (p[i] + 1), n[i][0] * n[i][1], 2 * p[i] * (p[i] + 1)) my_mesh = CrazyMesh( 2, n[i], ((-1, 1), (-1, 1)), curvature=curvature[i]) function_space = FunctionSpace(my_mesh, '1-lobatto', p[i]) form = BasisForm(function_space) form.quad_grid = 'gauss' form_1 = BasisForm(function_space) form_1.quad_grid = 'gauss' K = MeshFunction(my_mesh) K.continous_tensor = [diff_tens_11, diff_tens_12, diff_tens_22] M_1 = inner(form, form_1, K) for el in range(n[i][0] * n[i][1]): npt.assert_array_almost_equal( M_1_ref[:, el, :], M_1[:, :, el])
gamma = (gamma1, gamma2, gamma3, gamma4) dgamma = (dgamma1, dgamma2, dgamma3, dgamma4) ref_mesh = TransfiniteMesh(dim, elements_layout, gamma, dgamma) func_space_1_lobatto = FunctionSpace(ref_mesh, '1-lobatto', p, primal_is_inner) func_space_1_lobatto.dof_map.continous_dof = False func_space_0_ext_gauss = FunctionSpace(ref_mesh, '0-ext_gauss', p2, primal_is_inner) func_space_2_lobatto = FunctionSpace(ref_mesh, '2-lobatto', p, primal_is_inner) anisotropic_tensor = MeshFunction(ref_mesh) anisotropic_tensor.discrete_tensor = [k_11(), k_12(), k_22()] source_form = Form(func_space_2_lobatto) source_form.discretize(source) phi_0_exact = Form(func_space_0_ext_gauss) phi_0_exact.discretize(manufactured_solution) print(np.shape(phi_0_exact.cochain)) print(phi_0_exact.cochain) # define basis forms from function space basis_0 = BasisForm(func_space_0_ext_gauss) basis_1 = BasisForm(func_space_1_lobatto) basis_2 = BasisForm(func_space_2_lobatto)
def main(el, poly_degree): dim = 2 element_layout = (el + 1, el + 1) # print(element_layout) """define polynomial degree and inner/outer orientation""" pp = (poly_degree + 1, poly_degree + 1) # polynomial degree - primal mesh pd = (pp[0] - 1, pp[1] - 1) # polynomial degree - dual mesh orientation_inner = True outer = False # is_inner = False # orientation of primal mesh """define mesh""" bounds_domain = ((0, 1), (0, 1)) curvature = 0.0 mesh1 = CrazyMesh(dim, element_layout, bounds_domain, curvature) # gamma = (gamma1, gamma2, gamma3, gamma4) # dgamma = (dgamma1, dgamma2, dgamma3, dgamma4) # mesh1 = TransfiniteMesh(dim, element_layout, gamma, dgamma) """define function spaces used in problem""" fs_2_lobatto = FunctionSpace(mesh1, '2-lobatto', pp, outer) fs_1_lobatto = FunctionSpace(mesh1, '1-lobatto', pp, outer) fs_0_gauss = FunctionSpace(mesh1, '0-gauss', pd, inner) fs_1_lobatto.dof_map.continous_dof = True # continuous elements """define forms and quad grid""" """define (n) - source form""" f_source = Form(fs_2_lobatto) # form for source term f_source.discretize(source) f_source.basis.quad_grid = 'gauss' """define (n-1) - q form""" f_flux = Form(fs_1_lobatto) # form for flux terms f_flux.basis.quad_grid = 'lobatto' """define exact 0 - \phi form""" f_phi_exact = Form(fs_0_gauss) f_phi_exact.discretize(manufactured_solution) f_phi_exact.basis.quad_grid = 'gauss' """define unkown 0 - \phi form""" f_phi = Form(fs_0_gauss) f_phi.basis.quad_grid = 'gauss' """define anisotropic tensor as a mesh property""" # anisotropic_tensor = MeshFunction(mesh1) # anisotropic_tensor.discrete_tensor = [ # k_11(element_layout), k_12(element_layout), k_22(element_layout)] # mesh function to inject the anisotropic tensor anisotropic_tensor = MeshFunction(mesh1) anisotropic_tensor.continous_tensor = [k_11, k_12, k_22] # mesh_k = MeshFunction(crazy_mesh) # mesh_k.continous_tensor = [diffusion_11, diffusion_12, diffusion_22] """define basis functions""" basis_2 = BasisForm(fs_2_lobatto) basis_1 = BasisForm(fs_1_lobatto) basis_0 = BasisForm(fs_0_gauss) basis_2.quad_grid = 'gauss' basis_1.quad_grid = 'lobatto' basis_0.quad_grid = 'gauss' """general variables used frequently""" num_total_elements = element_layout[0] * element_layout[1] num_total_edges = fs_1_lobatto.num_dof num_total_faces = fs_2_lobatto.num_dof num_local_surfaces = fs_2_lobatto.num_local_dof dof_map_lobatto_faces = fs_2_lobatto.dof_map.dof_map """define 1-form mass matrix""" M1 = inner(basis_1, basis_1, anisotropic_tensor) M1_assembled = assemble(M1, (fs_1_lobatto, fs_1_lobatto)) """define the wedge product""" E21 = d_21_lobatto_outer(pp) W1 = basis_2.wedged(basis_0) W1_E21 = np.dot(W1, E21) W1_E21_local = np.repeat(W1_E21[:, :, np.newaxis], num_total_elements, axis=2) W1_E21_assembled = assemble(W1_E21_local, (fs_0_gauss, fs_1_lobatto)) """assemble lhs""" lhs = sparse.bmat([[M1_assembled, W1_E21_assembled.transpose()], [W1_E21_assembled, None]]).tolil() # A = np.linalg.det(lhs.todense()) # print(A) """assemble rhs""" rhs1 = np.zeros(num_total_edges)[:, np.newaxis] f_cochain_local = f_source.cochain_local[:, np.newaxis] W1_f_local = np.tensordot(W1, f_cochain_local, axes=1) rhs2 = assemble_cochain2(W1_f_local, dof_map_lobatto_faces, num_total_faces) rhs = np.vstack((rhs1, rhs2)) # print(time.time() - start_time) """implement boundary conditions""" """neuman boundary condition""" """dirichlet boundary condition""" """solve linear system of equations""" solution = sparse.linalg.spsolve(lhs.tocsc(), rhs) end_time = time.time() print("The total time taken by the program is : ", end_time - start_time) """l2 error""" """post processing / reconstruction""" eta_plot = xi_plot = xi = eta = np.linspace(-1, 1, 30) """reconstruct fluxes""" f_flux.cochain = solution[:fs_1_lobatto.num_dof] f_flux.reconstruct(xi, eta) (x_plot, y_plot), flux_x_plot, flux_y_plot = f_flux.export_to_plot() flux_x_plot, flux_y_plot = flux_y_plot, flux_x_plot """reconstruct potential""" f_phi.cochain = solution[fs_1_lobatto.num_dof:] f_phi.reconstruct(xi, eta) (x_plot, y_plot), phi_plot = f_phi.export_to_plot() phi_exact_plot = np.sin(2 * np.pi * x_plot) * np.sin(2 * np.pi * y_plot) """l2 - error in (div u -f)""" div_u_sum = np.zeros(num_total_elements) for ele_num in range(num_total_elements): l2_div_u = np.dot(E21, f_flux.cochain_local[:, ele_num] )[:, np.newaxis] - f_cochain_local[:, :, ele_num] div_u_sum[ele_num] = np.sum(l2_div_u) l2_err_div_u = np.linalg.norm(div_u_sum) l_inf_err_div_u = np.max(div_u_sum) """l2 - error in phi and flux """ l2_err_phi = f_phi.l_2_norm(phi_exact) l2_err_flux = f_flux.l_2_norm((flux_y_exact, flux_x_exact)) error = l2_err_phi, l2_err_flux, l2_err_div_u, l_inf_err_div_u print(l2_err_phi[0]) print(l2_err_flux[0]) # return error # plt.figure(1) plt.contourf(x_plot, y_plot, flux_x_plot) plt.colorbar() # # plt.figure(2) # plt.contourf(x_plot, y_plot, flux_x_exact_plot) # plt.colorbar() # print(np.max(flux_x_exact_plot), np.min(flux_x_exact_plot)) # print(np.max(flux_x_plot), np.min(flux_x_plot)) # plt.figure(3) plt.contourf(x_plot, y_plot, flux_y_plot) plt.colorbar() # # plt.figure(4) # plt.contourf(x_plot, y_plot, flux_y_exact_plot) # plt.colorbar() # print(np.max(flux_y_exact_plot), np.min(flux_y_exact_plot)) # print(np.max(flux_y_plot), np.min(flux_y_plot)) # plt.figure(5) plt.contourf(x_plot, y_plot, phi_plot) plt.colorbar() # plt.figure(6) # plt.contourf(x_plot, y_plot, phi_exact_plot) # plt.colorbar() # print(np.max(phi_exact_plot), np.min(phi_exact_plot)) # print(np.max(phi_plot), np.min(phi_plot)) plt.show()
row_idx = [value - 1 for value in row_idx] column_idx = [value - 1 for value in column_idx] data = 10**-6 * np.ones(np.size(row_idx)) k_11 = coo_matrix((data, (row_idx, column_idx)), shape=(20, 20)).toarray().ravel('F') k_11[np.where(k_11 < 10**-12)] = 1 return k_11.reshape(1, 400) def k_12(): return np.zeros((1, 400)) def k_22(): return k_11() anisotropic_tensor = MeshFunction(sand_shale_mesh) anisotropic_tensor.discrete_tensor = [k_11(), k_12(), k_22()] # define source term def source(x, y): return np.zeros(np.shape(x)) form_source = Form(func_space_2_lobatto) form_source.discretize(source) # define basis forms basis_1 = BasisForm(func_space_1_lobatto)