def test_full_mesh_refine():
knots = np.array([[0, 0, 1, 1], [0, 0, 1, 1]], dtype=np.float64)
deg = [1, 1]
dim = 2
T = HierarchicalSpace(knots, deg, dim)
np.testing.assert_equal(T.nfuncs_level, {0: 4})
cells = {}
rectangle = np.array([[0, 1 + np.spacing(1)], [0, 1 + np.spacing(1)]],
dtype=np.float64)
cells[0] = T.refine_in_rectangle(rectangle, 0)
T = refine(T, cells)
np.testing.assert_equal(T.nfuncs_level, {
0: 0,
1: 9,
})
cells[1] = T.refine_in_rectangle(rectangle, 1)
T = refine(T, cells)
np.testing.assert_equal(T.nfuncs_level, {0: 0, 1: 0, 2: 25})
cells[2] = T.refine_in_rectangle(rectangle, 2)
T = refine(T, cells)
np.testing.assert_equal(T.nfuncs_level, {0: 0, 1: 0, 2: 0, 3: 81})
def refine(hspace: HierarchicalSpace,
marked_entities,
type='cells') -> HierarchicalSpace:
"""
Refines the hierarchical space based on the type of marked entities.
:param hspace: hierarchical space
:param marked_entities: indices of the marked entities, in this case, cells
:param type: defaults to cells, not implemented 'function'-refinement yet.
:return: a refined hierarchical space
"""
logging.info("""
Refining hierarchical space
===========================
space.nlevels = {}
hmesh.nlevels = {}
""".format(hspace.nlevels, hspace.mesh.nlevels))
marked_cells = marked_entities
new_cells = hspace.mesh.refine(marked_cells)
marked_functions = hspace.functions_to_deactivate_from_cells(marked_cells)
hspace.refine(marked_functions, new_cells)
logging.info("""
After refinement, the space has
===============================
space.nfuncs = {}
space.nelems = {}
""".format(hspace.nfuncs, hspace.mesh.nel))
return hspace
def refine(hspace: HierarchicalSpace,
marked_entities,
type='cells') -> HierarchicalSpace:
"""
Refines the hierarchical space based on the type of marked entities.
:param hspace:
:param marked_entities:
:param type:
:return:
"""
logging.info("""
Refining hierarchical space
===========================
space.nlevels = {}
hmesh.nlevels = {}
""".format(hspace.nlevels, hspace.mesh.nlevels))
marked_cells = marked_entities
new_cells = hspace.mesh.refine(marked_cells)
marked_functions = hspace.functions_to_deactivate_from_cells(marked_cells)
hspace.refine(marked_functions, new_cells)
logging.info("""
After refinement, the space has
===============================
space.nfuncs = {}
space.nelems = {}
""".format(hspace.nfuncs, hspace.mesh.nel))
return hspace
def test_projection_matrix_bilinear():
knots = [[0, 0, 1, 2, 3, 3], [0, 0, 1, 2, 3, 3]]
d = [1, 1]
dim = 2
T = HierarchicalSpace(knots, d, dim)
cells = {0: [1]}
T = refine(T, cells)
C = T.compute_full_projection_matrix(0)
assert C.shape == (49, 16)
def test_functions_to_deactivate_from_cells():
knots = [[0, 0, 1, 2, 2], [0, 0, 1, 2, 2]]
d = 2
degrees = [1, 1]
T = HierarchicalSpace(knots, degrees, d)
marked_cells = {0: [0]}
new_cells = T.mesh.refine(marked_cells)
marked_functions = T.functions_to_deactivate_from_cells(marked_cells)
np.testing.assert_equal(marked_functions, {0: [0]})
def test_change_of_basis_matrix_linear():
knots = [[0, 0, 1 / 3, 2 / 3, 1, 1]]
d = [1]
dim = 1
T = HierarchicalSpace(knots, d, dim)
cells = {0: [1]}
T = refine(T, cells)
C = T.get_basis_conversion_matrix(0)
np.testing.assert_allclose(
C.toarray(),
np.array([[1, 0, 0, 0], [0.5, 0.5, 0, 0], [0, 1, 0, 0], [0, 0, 0, 0],
[0, 0, 1, 0], [0, 0, 0.5, 0.5], [0, 0, 0, 1]]))
def test_subdivision_matrix_linear():
knots = [[0, 0, 1 / 3, 2 / 3, 1, 1]]
d = [1]
dim = 1
T = HierarchicalSpace(knots, d, dim)
cells = {0: [1]}
T = refine(T, cells)
C = T.create_subdivision_matrix('full')
np.testing.assert_allclose(C[0].toarray(), np.eye(4))
np.testing.assert_allclose(
C[1].toarray(),
np.array([[1, 0, 0, 0, 0], [0.5, 0.5, 0, 0, 0], [0, 1, 0, 0, 0],
[0, 0, 0, 0, 1], [0, 0, 1, 0, 0], [0, 0, 0.5, 0.5, 0],
[0, 0, 0, 1, 0]]))
def test_invariant_refinement():
knots = [
[0] * 4 + [1 / 3, 2 / 3] + [1] * 4,
[0] * 4 + [1 / 3, 2 / 3] + [1] * 4,
]
deg = [3, 3]
dim = 2
t = HierarchicalSpace(knots, deg, dim)
m = hierarchical_mass_matrix(t).toarray()
# symmetry
np.testing.assert_allclose(m, m.T)
# partition of unity
np.testing.assert_allclose(np.sum(m, axis=(0, 1)), np.ones(m.shape[1]))
cells = {0: [0, 1]}
t = refine(t, cells)
m = hierarchical_mass_matrix(t).toarray()
# symmetry
np.testing.assert_allclose(m, m.T)
# partition of unity
np.testing.assert_allclose(np.sum(m, axis=(0, 1)), np.ones(m.shape[1]))
cells[1] = [0, 1, 2, 3]
t = refine(t, cells)
m = hierarchical_mass_matrix(t).toarray()
# symmetry
np.testing.assert_allclose(m, m.T)
# partition of unity
np.testing.assert_allclose(np.sum(m, axis=(0, 1)), np.ones(m.shape[1]))
def test_stiffness_matrix_subregion():
knots = [[0, 0, 0, 0, 1, 1, 1, 1], [0, 0, 0, 0, 1, 1, 1, 1]]
domain = np.array([[0, 1.0], [0, 1]])
deg = [3, 3]
dim = 2
T = HierarchicalSpace(knots, deg, dim)
m = hierarchical_mass_matrix(T).toarray().ravel()
m_local = local_mass_matrix(T, level=0,
element_indices=[0]).toarray().ravel()
a = hierarchical_stiffness_matrix(T).toarray().ravel()
a_local = local_stiffness_matrix(T, level=0,
element_indices=[0]).toarray().ravel()
assert np.all(np.in1d(m_local, m))
assert np.all(np.in1d(a_local, a))
m = hierarchical_mass_matrix(T).toarray().ravel()
m_local = local_mass_matrix(T, level=0,
element_indices=[0]).toarray().ravel()
a = hierarchical_stiffness_matrix(T).toarray().ravel()
a_local = local_stiffness_matrix(T, level=0,
element_indices=[0]).toarray().ravel()
assert np.all(np.in1d(m_local, m))
assert np.all(np.in1d(a_local, a))
def test_active_funcs_per_level():
knots = [[0, 0, 1, 1], [0, 0, 1, 1]]
d = 2
degrees = [1, 1]
T = HierarchicalSpace(knots, degrees, d)
np.testing.assert_equal(T.afunc_level, {0: [0, 1, 2, 3]})
def test_partition_of_unity():
knots = [[0, 0, 0, 1, 2, 3, 3, 3], [0, 0, 0, 1, 2, 3, 3, 3]]
d = 2
degrees = [2, 2]
T = HierarchicalSpace(knots, degrees, d)
marked_cells = {0: [0, 1, 2, 3]}
T = refine(T, marked_cells)
marked_cells = {0: [0, 1, 2, 3], 1: [0, 1, 2]}
T = refine(T, marked_cells)
C = create_subdivision_matrix(T)
N = 5
x = np.linspace(0, 3, N)
y = np.linspace(0, 3, N)
z = np.zeros((N, N))
c = C[T.nlevels - 1]
c = c.toarray()
for i in range(T.nfuncs):
u = np.zeros(T.nfuncs)
u[i] = 1
u_fine = c @ u
f = T.spaces[T.nlevels - 1].construct_function(u_fine)
for i in range(N):
for j in range(N):
z[i, j] += f(np.array([x[i], y[j]]))
np.testing.assert_allclose(z, 1)
def test_projection_matrix_linear():
knots = [[0, 0, 1, 2, 3, 3]]
d = [1]
dim = 1
T = HierarchicalSpace(knots, d, dim)
cells = {0: [1]}
T = refine(T, cells)
C = T.compute_full_projection_matrix(0)
assert C.shape == (7, 4)
np.testing.assert_allclose(
C.toarray(),
np.array([[1, 0, 0, 0], [0.5, 0.5, 0, 0], [0, 1, 0,
0], [0, 0.5, 0.5, 0],
[0, 0, 1, 0], [0, 0, 0.5, 0.5], [0, 0, 0, 1]]))
def test_linear_mass_matrix():
knots = [[0, 0, 1 / 3, 2 / 3, 1, 1]]
deg = [1]
dim = 1
T = HierarchicalSpace(knots, deg, dim)
cells = {0: [1]}
T = refine(T, cells)
M = hierarchical_mass_matrix(T)
assert T.nfuncs_level == {0: 4, 1: 1}
assert T.nfuncs == 5
assert M.shape == (5, 5)
expected_M = np.array([[
0.11111111,
0.05555556,
0.,
0.,
0.,
], [0.05555556, 0.16666667, 0., 0, 0.02777778],
[0., 0., 0.16666667, 0.05555556, 0.02777778],
[0., 0., 0.05555556, 0.11111111, 0.],
[0., 0.02777778, 0.02777778, 0., 0.11111111]])
np.testing.assert_allclose(M.toarray(), expected_M)
def test_stiffness_matrix_single_basis_func():
knots = [[0, 1, 2], [0, 1, 2]]
deg = [1, 1]
dim = 2
T = HierarchicalSpace(knots, deg, dim)
a = hierarchical_stiffness_matrix(T).toarray()
np.testing.assert_allclose(a, 8 / 3)
def test_active_funcs_per_level_refine():
knots = [[0, 0, 1, 2, 2], [0, 0, 1, 2, 2]]
d = 2
degrees = [1, 1]
T = HierarchicalSpace(knots, degrees, d)
cells = {0: [0]}
T = refine(T, cells)
np.testing.assert_equal(T.nfuncs_level, {0: 8, 1: 4})
def test_local_mass_matrix_univariate():
knots = [[0, 1, 2]]
deg = [1]
dim = 1
T = HierarchicalSpace(knots, deg, dim)
M = local_mass_matrix(T, 0)
assert M.shape == (1, 1)
np.testing.assert_allclose(M.toarray(), 2 / 3)
def test_local_mass_matrix_univariate_refined():
knots = [[0, 0, 1 / 3, 2 / 3, 1, 1]]
deg = [1]
dim = 1
T = HierarchicalSpace(knots, deg, dim)
cell = {0: [1]}
T = refine(T, cell)
m0 = local_mass_matrix(T, 0)
m1 = local_mass_matrix(T, 1)
assert m0.shape == (4, 4)
assert m1.shape == (7, 7)
def create_subdivision_matrix(hspace: HierarchicalSpace) -> dict:
"""
Returns hspace.nlevels-1 matrices used for representing coarse B-splines in terms of the finer B-splines.
:param hspace: HierarchicalSpace containing the needed information
:return: a dictionary mapping
"""
mesh = hspace.mesh
C = {}
C[0] = sp.identity(hspace.spaces[0].nfuncs, format='lil')
C[0] = C[0][:, hspace.afunc_level[0]]
for level in range(1, hspace.nlevels):
I = sp.identity(hspace.spaces[level].nfuncs, format='lil')
aux = sp.lil_matrix(hspace.get_basis_conversion_matrix(level - 1))
C[level] = sp.hstack(
[aux @ C[level - 1], I[:, hspace.afunc_level[level]]])
return C
Refining hierarchical space
===========================
space.nlevels = {}
hmesh.nlevels = {}
""".format(hspace.nlevels, hspace.mesh.nlevels))
marked_cells = marked_entities
new_cells = hspace.mesh.refine(marked_cells)
marked_functions = hspace.functions_to_deactivate_from_cells(marked_cells)
hspace.refine(marked_functions, new_cells)
logging.info("""
After refinement, the space has
===============================
space.nfuncs = {}
space.nelems = {}
""".format(hspace.nfuncs, hspace.mesh.nel))
return hspace
if __name__ == '__main__':
knots = [[0, 0, 1, 2, 3, 3], [0, 0, 1, 2, 3, 3]]
d = 2
degrees = [1, 1]
T = HierarchicalSpace(knots, degrees, d)
marked_cells = {0: [0, 1, 2, 3, 4, 5, 6]}
T = refine(T, marked_cells)
marked_cells = {0: [0, 1, 2, 3, 4, 5, 6], 1: [0, 1, 2, 3, 4, 5, 6]}
T = refine(T, marked_cells)
T.mesh.plot_cells()
print(T)
def test_bilinear_mass_and_stiffness_matrix():
knots = [[0, 0, 1 / 3, 2 / 3, 1, 1], [0, 0, 1 / 3, 2 / 3, 1, 1]]
deg = [1, 1]
dim = 2
t = HierarchicalSpace(knots, deg, dim)
cells = {0: [1]}
t = refine(t, cells)
m = hierarchical_mass_matrix(t)
expected_m = np.array([
[
0.012346, 0.006173, 0.000000, 0.000000, 0.006173, 0.003086,
0.000000, 0.000000, 0.000000, 0.000000, 0.000000, 0.000000,
0.000000, 0.000000, 0.000000, 0.000000, 0.000000, 0.000000
],
[
0.006173, 0.018519, 0.000000, 0.000000, 0.003086, 0.009452,
0.000193, 0.000000, 0.000000, 0.000000, 0.000000, 0.000000,
0.000000, 0.000000, 0.000000, 0.000000, 0.001929, 0.002315
],
[
0.000000, 0.000000, 0.018519, 0.006173, 0.000000, 0.000193,
0.009452, 0.003086, 0.000000, 0.000000, 0.000000, 0.000000,
0.000000, 0.000000, 0.000000, 0.000000, 0.001929, 0.002315
],
[
0.000000, 0.000000, 0.006173, 0.012346, 0.000000, 0.000000,
0.003086, 0.006173, 0.000000, 0.000000, 0.000000, 0.000000,
0.000000, 0.000000, 0.000000, 0.000000, 0.000000, 0.000000
],
[
0.006173, 0.003086, 0.000000, 0.000000, 0.024691, 0.012346,
0.000000, 0.000000, 0.006173, 0.003086, 0.000000, 0.000000,
0.000000, 0.000000, 0.000000, 0.000000, 0.000000, 0.000000
],
[
0.003086, 0.009452, 0.000193, 0.000000, 0.012346, 0.046682,
0.009645, 0.000000, 0.003086, 0.012346, 0.003086, 0.000000,
0.000000, 0.000000, 0.000000, 0.000000, 0.000386, 0.003858
],
[
0.000000, 0.000193, 0.009452, 0.003086, 0.000000, 0.009645,
0.046682, 0.012346, 0.000000, 0.003086, 0.012346, 0.003086,
0.000000, 0.000000, 0.000000, 0.000000, 0.000386, 0.003858
],
[
0.000000, 0.000000, 0.003086, 0.006173, 0.000000, 0.000000,
0.012346, 0.024691, 0.000000, 0.000000, 0.003086, 0.006173,
0.000000, 0.000000, 0.000000, 0.000000, 0.000000, 0.000000
],
[
0.000000, 0.000000, 0.000000, 0.000000, 0.006173, 0.003086,
0.000000, 0.000000, 0.024691, 0.012346, 0.000000, 0.000000,
0.006173, 0.003086, 0.000000, 0.000000, 0.000000, 0.000000
],
[
0.000000, 0.000000, 0.000000, 0.000000, 0.003086, 0.012346,
0.003086, 0.000000, 0.012346, 0.049383, 0.012346, 0.000000,
0.003086, 0.012346, 0.003086, 0.000000, 0.000000, 0.000000
],
[
0.000000, 0.000000, 0.000000, 0.000000, 0.000000, 0.003086,
0.012346, 0.003086, 0.000000, 0.012346, 0.049383, 0.012346,
0.000000, 0.003086, 0.012346, 0.003086, 0.000000, 0.000000
],
[
0.000000, 0.000000, 0.000000, 0.000000, 0.000000, 0.000000,
0.003086, 0.006173, 0.000000, 0.000000, 0.012346, 0.024691,
0.000000, 0.000000, 0.003086, 0.006173, 0.000000, 0.000000
],
[
0.000000, 0.000000, 0.000000, 0.000000, 0.000000, 0.000000,
0.000000, 0.000000, 0.006173, 0.003086, 0.000000, 0.000000,
0.012346, 0.006173, 0.000000, 0.000000, 0.000000, 0.000000
],
[
0.000000, 0.000000, 0.000000, 0.000000, 0.000000, 0.000000,
0.000000, 0.000000, 0.003086, 0.012346, 0.003086, 0.000000,
0.006173, 0.024691, 0.006173, 0.000000, 0.000000, 0.000000
],
[
0.000000, 0.000000, 0.000000, 0.000000, 0.000000, 0.000000,
0.000000, 0.000000, 0.000000, 0.003086, 0.012346, 0.003086,
0.000000, 0.006173, 0.024691, 0.006173, 0.000000, 0.000000
],
[
.000000, 0.000000, 0.000000, 0.000000, 0.000000, 0.000000,
0.000000, 0.000000, 0.000000, 0.000000, 0.003086, 0.006173,
0.000000, 0.000000, 0.006173, 0.012346, 0.000000, 0.000000
],
[
.000000, 0.001929, 0.001929, 0.000000, 0.000000, 0.000386,
0.000386, 0.000000, 0.000000, 0.000000, 0.000000, 0.000000,
0.000000, 0.000000, 0.000000, 0.000000, 0.006173, 0.003086
],
[
.000000, 0.002315, 0.002315, 0.000000, 0.000000, 0.003858,
0.003858, 0.000000, 0.000000, 0.000000, 0.000000, 0.000000,
0.000000, 0.000000, 0.000000, 0.000000, 0.003086, 0.012346
],
])
np.testing.assert_allclose(m.toarray(), expected_m, atol=1.0e-5)