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()
def test_assembly_inner_product_2_forms(self):
"""Test the assembly of 1-forms inner products."""
func_space_lob = FunctionSpace(self.mesh, '2-lobatto', self.p)
func_space_gauss = FunctionSpace(self.mesh, '2-gauss', self.p)
func_space_extgauss = FunctionSpace(self.mesh, '2-ext_gauss', self.p)
basis_lob = BasisForm(func_space_lob)
basis_lob.quad_grid = 'gauss'
M_lob = inner(basis_lob, basis_lob)
basis_gauss = BasisForm(func_space_gauss)
basis_gauss.quad_grid = 'lobatto'
M_gauss = inner(basis_gauss, basis_gauss)
basis_ext_gauss = BasisForm(func_space_extgauss)
print(basis_ext_gauss.num_basis)
basis_ext_gauss.quad_grid = 'lobatto'
M_extgauss = inner(basis_ext_gauss, basis_ext_gauss)
M_lob_ass_ref = assemble_slow(self.mesh, M_lob, func_space_lob.dof_map.dof_map,
func_space_lob.dof_map.dof_map)
M_gauss_ass_ref = assemble_slow(self.mesh, M_gauss, func_space_gauss.dof_map.dof_map,
func_space_gauss.dof_map.dof_map)
M_extgauss_ass_ref = assemble_slow(
self.mesh, M_extgauss, func_space_extgauss.dof_map.dof_map_internal, func_space_extgauss.dof_map.dof_map_internal)
M_lob_ass = assemble(M_lob, func_space_lob, func_space_lob).toarray()
M_gauss_ass = assemble(M_gauss, func_space_gauss, func_space_gauss).toarray()
M_extgauss_ass = assemble(M_extgauss, func_space_extgauss,
func_space_extgauss).toarray()
npt.assert_array_almost_equal(M_lob_ass_ref, M_lob_ass)
npt.assert_array_almost_equal(M_gauss_ass_ref, M_gauss_ass)
npt.assert_array_almost_equal(M_extgauss_ass_ref, M_extgauss_ass)
def test_basis_input(self):
"""Test the coboundary operator with some basis as inputs."""
p_x, p_y = 2, 2
func_space_0 = FunctionSpace(self.mesh,
'0-lobatto', (p_x, p_y),
is_inner=False)
func_space_1 = FunctionSpace(self.mesh,
'1-lobatto', (p_x, p_y),
is_inner=False)
func_space_2 = FunctionSpace(self.mesh,
'2-lobatto', (p_x, p_y),
is_inner=False)
basis_0_ref = BasisForm(func_space_0)
basis_1_ref = BasisForm(func_space_1)
basis_0_ref.quad_grid = 'lobatto'
basis_1_ref.quad_grid = 'gauss'
basis_2_ref = BasisForm(func_space_2)
basis_2_ref.quad_grid = 'lobatto'
e_21_ref = d_21_lobatto_outer((p_x, p_y))
e_10_ref = d_10_lobatto_outer((p_x, p_y))
basis_1, e_10 = d(basis_0_ref)
basis_1.quad_grid = 'gauss'
basis_2, e_21 = d(basis_1_ref)
basis_2.quad_grid = 'lobatto'
M_1 = inner(basis_1, basis_1)
M_1_ref = inner(basis_1_ref, basis_1_ref)
npt.assert_array_almost_equal(M_1_ref, M_1)
M_2 = inner(basis_2, basis_2)
M_2_ref = inner(basis_2_ref, basis_2_ref)
npt.assert_array_almost_equal(M_2_ref, M_2)
npt.assert_array_equal(e_21_ref, e_21)
npt.assert_array_equal(e_10_ref, e_10)
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_value_face_basis(self):
for p in range(2, 15):
function_space = FunctionSpace(self.mesh, '2-lobatto', p)
basis = BasisForm(function_space)
basis.quad_grid = 'lobatto'
ref_funcs = np.loadtxt(
os.getcwd() + '/src/tests/test_basis_2_form/basis_2_form_p_' +
str(p) + '.dat',
delimiter=',')
# decimals limited due to the limited precision of the matlab data
npt.assert_array_almost_equal(ref_funcs, basis.basis, decimal=1)
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_basis_inner(self):
"""Test inner product with basis functions."""
p_x, p_y = 2, 2
func_space_0 = FunctionSpace(self.mesh, '0-lobatto', (p_x, p_y))
func_space_1 = FunctionSpace(self.mesh, '1-lobatto', (p_x, p_y))
basis_0 = BasisForm(func_space_0)
basis_1 = BasisForm(func_space_1)
basis_1.quad_grid = 'lobatto'
basis_0.quad_grid = 'lobatto'
M_1 = inner(basis_1, basis_1)
e_10 = d(func_space_0)
inner_prod_ref = np.tensordot(e_10, M_1, axes=((0), (0)))
#
inner_prod = inner(d(basis_0), basis_1)
npt.assert_array_almost_equal(inner_prod_ref, inner_prod)
#
inner_prod_1_ref = np.rollaxis(
np.tensordot(M_1, e_10, axes=((0), (0))), 2)
inner_prod_1 = inner(basis_1, d(basis_0))
npt.assert_array_almost_equal(inner_prod_1_ref, inner_prod_1)
#
inner_prod_2_ref = np.tensordot(e_10,
np.tensordot(e_10,
M_1,
axes=((0), (0))),
axes=((0), (1)))
inner_prod_2 = inner(d(basis_0), d(basis_0))
#
npt.assert_array_almost_equal(inner_prod_2_ref, inner_prod_2)
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_inner(self):
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]):
print("Test case n : ", i)
# basis = basis_forms.
# print("theo size ", (p[i] + 1)**2 *
# (p[i] + 1)**2 * n[i][0] * n[i][1])
M_2_ref = np.loadtxt(
os.getcwd() + '/src/tests/test_M_2/M_2_' + case + '.dat',
delimiter=',').reshape((p[i])**2, n[i][0] * n[i][1], (p[i])**2)
# print("actual size ", np.size(M_0_ref))
# print(np.shape(M_0_ref))
my_mesh = CrazyMesh(2,
n[i], ((-1, 1), (-1, 1)),
curvature=curvature[i])
function_space = FunctionSpace(my_mesh, '2-lobatto', p[i])
basis_0 = BasisForm(function_space)
basis_0.quad_grid = 'lobatto'
basis_1 = BasisForm(function_space)
basis_1.quad_grid = 'lobatto'
M_2 = inner(basis_0, basis_1)
# print("REF ------------------ \n", M_2_ref[:, 0, :])
# print("CALCULATED _----------------\n", M_2[:, :, 0])
for el in range(n[i][0] * n[i][1]):
npt.assert_array_almost_equal(M_2_ref[:, el, :],
M_2[:, :, el],
decimal=7)
def test_inner(self):
list_cases = [
'p2_n2-2', 'p2_n3-2', 'p5_n1-10', 'p10_n2-2', 'p18_n14-14'
]
p = [2, 2, 5, 10, 18]
n = [(2, 2), (3, 2), (1, 10), (2, 2), (14, 10)]
curvature = [0.2, 0.1, 0.2, 0.2, 0.2]
for i, case in enumerate(list_cases[:-1]):
print("Test case n : ", i)
# basis = basis_forms.
# print("theo size ", (p[i] + 1)**2 *
# (p[i] + 1)**2 * n[i][0] * n[i][1])
M_0_ref = np.loadtxt(
os.getcwd() + '/src/tests/test_M_0/M_0_' + case + '.dat',
delimiter=',').reshape((p[i] + 1)**2, n[i][0] * n[i][1],
(p[i] + 1)**2)
# print(np.shape(M_0_ref))
my_mesh = CrazyMesh(2,
n[i], ((-1, 1), (-1, 1)),
curvature=curvature[i])
function_space = FunctionSpace(my_mesh, '0-lobatto', p[i])
basis = BasisForm(function_space)
basis.quad_grid = 'lobatto'
basis_1 = BasisForm(function_space)
basis_1.quad_grid = 'lobatto'
M_0 = inner(basis, basis_1)
for el in range(n[i][0] * n[i][1]):
npt.assert_array_almost_equal(M_0_ref[:, el, :],
M_0[:, :, el],
decimal=4)
def d(state):
"""Exterior derivative operator.
The function reppresent the discrete version of the exterior derivative:
the coboundary operator.
The function can act on:
-FunctionSpace,
-Form,
-BasisForm.
It returns respectively:
-the incidence matrix,
-a form of degree + 1, whose cochain is the result of matrix multiplication of the original cochain with the incidence matrix,
-a tuple made by the basisform, of 1 degree higher, and the incidence matrix.
"""
if isinstance(state, FunctionSpace):
incidence_matrix = fetch_incidence(state)
return incidence_matrix
if isinstance(state, AbstractForm):
function_space = state.function_space
incidence_matrix = fetch_incidence(function_space)
# generate the next func space in the de Rham sequence
d_function_space = NextSpace(function_space)
# calculate the new cochain
d_form = Form(d_function_space)
d_form.cochain_local = np.dot(incidence_matrix, state.cochain_local)
d_form.cochain_to_global()
return d_form
if isinstance(state, AbstractBasisForm):
function_space = state.function_space
incidence_matrix = fetch_incidence(function_space)
# generate the next func space in the de Rham sequence
d_function_space = NextSpace(function_space)
d_basis = BasisForm(d_function_space)
d_basis.quad_grid = parse_quadrature(state)
return (d_basis, incidence_matrix)
def test_exceptions(self):
p = 2
mesh = CrazyMesh(2, (2, 2), ((-1, 1), (-1, 1)))
func_space = FunctionSpace(mesh, '1-lobatto', p)
basis = BasisForm(func_space)
basis_1 = BasisForm(func_space)
quad_cases = [(None, 'gauss'), (None, None), ('gauss', None), ('lobatto', 'gauss')]
for case in quad_cases:
if None in case:
with self.assertRaises(UnboundLocalError):
basis.quad_grid = case[0]
basis_1.quad_grid = case[1]
else:
basis.quad_grid = case[0]
basis_1.quad_grid = case[1]
self.assertRaises(QuadratureError, inner, basis, basis_1)
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_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])
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()
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(), k_12(), k_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 = 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)
basis_2 = BasisForm(func_space_2_lobatto)
basis_1.quad_grid = 'lobatto'
basis_2.quad_grid = 'lobatto'
# solution form
phi_2 = Form(func_space_2_lobatto)
phi_2.basis.quad_grid = 'lobatto'
q_1 = Form(func_space_1_lobatto)
q_1.basis.quad_grid = 'lobatto'
form_source = Form(func_space_2_lobatto)
form_source.discretize(source)
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()
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) basis_0.quad_grid = 'gauss' basis_1.quad_grid = 'lobatto' basis_2.quad_grid = 'lobatto' # define forms for soultion using function space q_1 = Form(func_space_1_lobatto) # q_1.basis.quad_grid = 'lobatto' phi_0 = Form(func_space_0_ext_gauss) # phi_0.basis.quad_grid = 'gauss' E21 = d_21_lobatto_outer(p)
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()