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 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_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_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_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_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_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_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 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_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]]))
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
if __name__ == '__main__':
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)
C = create_subdivision_matrix(T)
N = 30
x = np.linspace(0, 3, N)
y = np.linspace(0, 3, N)
z = np.zeros((N, N))
X, Y = np.meshgrid(x, y)
B = T.spaces[-1].basis
c = C[T.nlevels - 1]
c = c.toarray()
for i in range(T.nfuncs):
u = np.zeros(T.nfuncs)
u[i] = 1
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)