def multiple_element():
mesh = CrazyMesh(2, (5, 7), ((-1, 1), (-1, 1)), curvature=0.3)
p = 10, 10
func_space_vort = FunctionSpace(mesh, '0-ext_gauss', (p[0] - 1, p[1] - 1), is_inner=False)
func_space_outer_vel = FunctionSpace(
mesh, '1-total_ext_gauss', (p[0], p[1]), is_inner=False)
# func_space_outer_vel = FunctionSpace(
# mesh, '1-gauss', (p[0], p[1]), is_inner=False)
func_space_inner_vel = FunctionSpace(mesh, '1-lobatto', p, is_inner=True)
func_space_inner_vel.dof_map.continous_dof = True
func_space_source = FunctionSpace(mesh, '2-lobatto', p, is_inner=True)
basis_vort = BasisForm(func_space_vort)
basis_vel_in = BasisForm(func_space_inner_vel)
basis_vel_in.quad_grid = 'lobatto'
basis_vel_out = BasisForm(func_space_outer_vel)
basis_vel_out.quad_grid = 'lobatto'
basis_2 = BasisForm(func_space_source)
psi = Form(func_space_vort)
u_in = Form(func_space_inner_vel)
source = Form(func_space_source)
source.discretize(ffun)
M_1 = inner(basis_vel_in, basis_vel_in)
E_21_in = d(func_space_inner_vel)
W_02 = basis_2.wedged(basis_vort)
W_02_E21 = np.transpose(W_02 @ E_21_in)
M_1 = assemble(M_1, (func_space_inner_vel, func_space_inner_vel))
print(np.shape(M_1))
W_02_E21 = assemble(W_02_E21, (func_space_inner_vel, func_space_vort))
print(np.shape(W_02_E21))
W_02 = assemble(W_02, (func_space_source, func_space_vort))
lhs = spr.bmat([[M_1, W_02_E21], [W_02_E21.transpose(), None]])
print(np.shape(lhs))
rhs = np.zeros(np.shape(lhs)[0])
rhs[-func_space_source.num_dof:] = W_02 @ source.cochain
solution = spr.linalg.spsolve(lhs.tocsc(), rhs)
u_in.cochain = solution[:func_space_inner_vel.num_dof]
cochian_psi = np.zeros(func_space_vort.num_dof)
cochian_psi[:func_space_vort.num_internal_dof] = solution[-func_space_vort.num_internal_dof:]
psi.cochain = cochian_psi
xi = eta = np.linspace(-1, 1, 40)
u_in.reconstruct(xi, eta)
(x, y), u_x, u_y = u_in.export_to_plot()
plt.contourf(x, y, u_x)
plt.colorbar()
plt.title("u_x inner")
plt.show()
psi.reconstruct(xi, eta)
(x, y), psi_value = psi.export_to_plot()
plt.contourf(x, y, psi_value)
plt.title("psi outer"
)
plt.colorbar()
plt.show()
edge_id = px * (px + 1) + j
virtual_E21[vol_id - px**2, edge_id] = 1
virtual_E21[vol_id - px**2 + px, edge_id + px**2] = -1
for i in range(px):
vol_id = px**2 + 2 * px + i
edge_id = (px + 1) * i
virtual_E21[vol_id - px**2, edge_id] = 1
virtual_E21[vol_id - px**2 + px, edge_id + px] = -1
# define the mass and wedge matrices
M_1 = inner(basis_1, basis_1)
M1_assembled = assemble(M_1, (func_space_1_lobatto, func_space_1_lobatto))
W_i_2 = basis_2.wedged(basis_0)
Wi_2_E_21 = np.dot(W_i_2, E21)
lhs = sparse.bmat(
[[M1_assembled,
Wi_2_E_21.transpose(),
virtual_E21.transpose()], [Wi_2_E_21, None, None],
[virtual_E21, None, None]]).tolil()
rhs_1 = np.zeros(2 * px * (px + 1))
rhs_2 = np.dot(W_i_2, source_form.cochain)
rhs_3 = np.zeros(4 * px)
rhs = np.hstack((rhs_1, rhs_2, rhs_3))
# implementing BC's
elements_layout[1] + 1) + py * elements_layout[1] * (elements_layout[0] +
1)
num_local_element_edges = 2 * px * (px + 1)
num_total_edges = func_space_1_lobatto.num_dof
num_total_surfaces = func_space_2_lobatto.num_dof
num_local_surfaces = func_space_2_lobatto.num_local_dof
dof_map_ext_gauss_nodes = func_space_0_ext_gauss.dof_map.dof_map
dof_map_lobatto_edges_discontinous = func_space_1_lobatto.dof_map.dof_map
dof_map_lobatto_faces = dof_map_ext_gauss_nodes
"""Define 1-form mass matrix"""
M_1 = inner(basis_1, basis_1)
M1_assembled = assemble(M_1, (func_space_1_lobatto, func_space_1_lobatto))
"""Define wedge 1"""
E21 = d_21_lobatto_outer(p)
W1 = basis_2.wedged(basis_0)
W1_E21 = np.dot(W1, E21)
W1_E21_3D = np.repeat(W1_E21[:, :, np.newaxis], num_total_elements, axis=2)
W1_E21_assembled = assemble(W1_E21_3D,
(func_space_0_ext_gauss, func_space_1_lobatto))
"""Define Wedge 2"""
"""Define the second wedge / duality pairing / element connectivity matrix"""
virtual_E21 = d_21_lobatto_outer_virtual(p)
W2 = wedge_2()
W2_vE21 = np.dot(W2, virtual_E21)
W2_vE21_3D = np.repeat(W2_vE21[:, :, np.newaxis], num_total_elements, axis=2)
W2_vE21_assembled = assemble_virtual_E21(W2_vE21_3D)
def multiple_element_v1():
mesh = CrazyMesh(2, (1, 1), ((-1, 1), (-1, 1)), curvature=0.0)
p = 4, 4
func_space_vort = FunctionSpace(mesh, '0-lobatto', p, is_inner=False)
func_space_outer_vel = FunctionSpace(
mesh, '1-lobatto', p, is_inner=False)
func_space_outer_vel.dof_map.continous_dof = True
# func_space_outer_vel = FunctionSpace(
# mesh, '1-gauss', (p[0], p[1]), is_inner=False)
func_space_inner_vel = FunctionSpace(
mesh, '1-gauss', (p[0], p[1]), is_inner=True)
print('dof gauss :', func_space_inner_vel.num_dof)
func_space_inner_vel.dof_map.continous_dof = False
func_space_source = FunctionSpace(mesh, '2-gauss', (p[0] + 1, p[1] + 1), is_inner=True)
print("dof source :", func_space_source.num_dof)
basis_vort = BasisForm(func_space_vort)
basis_vel_in = BasisForm(func_space_inner_vel)
basis_vel_in.quad_grid = 'lobatto'
basis_vel_out = BasisForm(func_space_outer_vel)
basis_vel_out.quad_grid = 'lobatto'
basis_2 = BasisForm(func_space_source)
psi = Form(func_space_vort)
u_in = Form(func_space_inner_vel)
source = Form(func_space_source)
source.discretize(ffun)
M_1 = inner(basis_vel_in, basis_vel_in)
# E_21_in = d(func_space_inner_vel)
W_02 = basis_2.wedged(basis_vort)
W_11 = basis_vel_in.wedged(basis_vel_out)
E_10_out = d(func_space_vort)
print(np.shape(W_11), np.shape(E_10_out))
W_E = W_11 @ E_10_out
W_11_inv = basis_vel_out.wedged(basis_vel_in)
E_W = np.transpose(E_10_out) @ W_11_inv
print("shape ew : ", np.shape(W_E))
print(np.shape(W_02))
M_1 = assemble(M_1, (func_space_inner_vel, func_space_inner_vel))
print(func_space_source.num_local_dof)
W_E = assemble(W_E, (func_space_outer_vel, func_space_vort))
E_W = assemble(E_W, (func_space_vort, func_space_outer_vel))
W_02 = assemble(W_02, (func_space_vort, func_space_source))
lhs = spr.bmat([[M_1, W_E], [W_E.transpose(), None]])
rhs = np.zeros(np.shape(lhs)[0])
rhs[-func_space_source.num_dof:] = W_02 @ source.cochain
solution = spr.linalg.spsolve(lhs.tocsc(), rhs)
u_in.cochain = solution[:func_space_inner_vel.num_dof]
psi.cochain = solution[-func_space_vort.num_dof:]
xi = eta = np.linspace(-1, 1, 40)
u_in.reconstruct(xi, eta)
(x, y), u_x, u_y = u_in.export_to_plot()
plt.contourf(x, y, u_x)
plt.colorbar()
plt.title("u_x inner")
plt.show()
psi.reconstruct(xi, eta)
(x, y), psi_value = psi.export_to_plot()
plt.contourf(x, y, psi_value)
plt.title("psi outer")
plt.colorbar()
plt.show()
def single_element():
mesh = CrazyMesh(2, (1, 1), ((-1, 1), (-1, 1)), curvature=0.1)
p = 20, 20
func_space_vort = FunctionSpace(mesh, '0-ext_gauss', (p[0] - 1, p[1] - 1), is_inner=False)
func_space_outer_vel = FunctionSpace(
mesh, '1-ext_gauss', (p[0] - 2, p[1] - 2), is_inner=False)
func_space_inner_vel = FunctionSpace(mesh, '1-lobatto', p, is_inner=True)
func_space_source = FunctionSpace(mesh, '2-lobatto', p, is_inner=True)
basis_vort = BasisForm(func_space_vort)
basis_vel_in = BasisForm(func_space_inner_vel)
basis_vel_out = BasisForm(func_space_outer_vel)
basis_2 = BasisForm(func_space_source)
psi = Form(func_space_vort)
u_in = Form(func_space_inner_vel)
source = Form(func_space_source)
source.discretize(ffun)
M_1 = inner(basis_vel_in, basis_vel_in)
W_11 = basis_vel_out.wedged(basis_vel_in)
E_10_ext = d(func_space_vort)
E_21_in = d(func_space_inner_vel)
W_02 = basis_vort.wedged(basis_2)
print("shapes : \n \
M_1 : {0} \n \
W_11 : {1} \n \
E_10 : {2} \n \
E_21_in : {3} \n \
W_02 : {4} \n" .format(np.shape(M_1), np.shape(W_11), np.shape(E_10_ext), np.shape(E_21_in), np.shape(W_02)))
# one element
# print(func_space_inner_vel.num_dof, func_space_vort.num_dof)
lhs_0 = np.hstack((M_1[:, :, 0], np.transpose(W_02 @ E_21_in)))
col_size_0 = np.shape(lhs_0)[1]
eW = W_02 @ E_21_in
col_zeros = col_size_0 - np.shape(eW)[1]
lhs_1 = np.hstack((eW, np.zeros(
(func_space_source.num_dof, col_zeros))))
lhs = np.vstack((lhs_0, lhs_1))
rhs_source = (source.cochain @ W_02)[:, np.newaxis]
rhs_zeros = np.zeros((np.shape(lhs)[0] - func_space_source.num_dof, 1))
rhs = np.vstack((rhs_zeros, rhs_source))
print(np.shape(lhs))
solution = np.linalg.solve(lhs, rhs).flatten()
print(np.shape(solution))
print(func_space_vort.num_dof)
u_in.cochain = solution[:func_space_inner_vel.num_dof]
psi_zeros = np.zeros((func_space_vort.num_dof - func_space_vort.num_internal_local_dof))
psi.cochain = np.append(solution[func_space_inner_vel.num_dof:],
np.zeros((func_space_vort.num_dof - func_space_vort.num_internal_local_dof)))
xi = eta = np.linspace(-1, 1, 40)
u_in.reconstruct(xi, eta)
(x, y), u_x, u_y = u_in.export_to_plot()
plt.contourf(x, y, u_x)
plt.colorbar()
plt.title("u_x inner")
plt.show()
psi.reconstruct(xi, eta)
(x, y), psi_value = psi.export_to_plot()
plt.contourf(x, y, psi_value)
plt.title("psi outer"
)
plt.colorbar()
plt.show()
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()
