def test_tensor_div_div(conservative):
"""test double divergence of a tensor field by comparison with two divergences"""
grid = SphericalSymGrid([0, 1], 64)
expr = "r * tanh((0.5 - r) * 10)"
bc = "auto_periodic_neumann"
# test radial part
tf = Tensor2Field.from_expression(grid,
[[expr, 0, 0], [0, 0, 0], [0, 0, 0]])
res = tf._apply_operator("tensor_double_divergence",
bc=bc,
conservative=conservative)
est = tf.divergence(bc).divergence(bc)
np.testing.assert_allclose(res.data[2:-2],
est.data[2:-2],
rtol=0.02,
atol=1)
# test angular part
tf = Tensor2Field.from_expression(grid,
[[0, 0, 0], [0, expr, 0], [0, 0, expr]])
res = tf._apply_operator("tensor_double_divergence",
bc=bc,
conservative=conservative)
est = tf.divergence(bc).divergence(bc)
np.testing.assert_allclose(res.data[2:-2],
est.data[2:-2],
rtol=0.02,
atol=1)
def test_tensor_div_div_analytical():
"""test double divergence of a tensor field against analytical expression"""
grid = SphericalSymGrid([0.5, 1], 12)
tf = Tensor2Field.from_expression(
grid, [["r**4", 0, 0], [0, "r**3", 0], [0, 0, "r**3"]])
res = tf._apply_operator("tensor_double_divergence", bc="curvature")
expect = ScalarField.from_expression(grid, "2 * r * (15 * r - 4)")
np.testing.assert_allclose(res.data[1:-1], expect.data[1:-1], rtol=0.01)
def test_examples_tensor_polar():
"""compare derivatives of tensorial fields for polar grids"""
grid = PolarSymGrid(1, 32)
tf = Tensor2Field.from_expression(grid, [["r**3"] * 2] * 2)
# tensor divergence
res = tf.divergence([{"derivative_normal": 0}, {"value_normal": [1, 1]}])
expect = VectorField.from_expression(grid, ["3 * r**2", "5 * r**2"])
np.testing.assert_allclose(res.data, expect.data, rtol=0.1, atol=0.1)
def test_examples_tensor_sph():
"""compare derivatives of tensorial fields for spherical grids"""
grid = SphericalSymGrid(1, 32)
tf = Tensor2Field.from_expression(grid, [["r**3"] * 3] * 3)
tfd = tf.data
tfd[0, 1] = tfd[1, 1] = tfd[1, 2] = tfd[2, 1] = tfd[2, 2] = 0
# tensor divergence
res = tf.divergence([{"derivative_normal": 0}, {"value_normal": [1, 1, 1]}])
expect = VectorField.from_expression(grid, ["5 * r**2", "5 * r**2", "6 * r**2"])
np.testing.assert_allclose(res.data, expect.data, rtol=0.1, atol=0.1)
def test_from_expressions():
"""test initializing tensor fields with expressions"""
grid = UnitGrid([4, 4])
tf = Tensor2Field.from_expression(grid, [[1, 1], ["x**2", "x * y"]])
xs = grid.cell_coords[..., 0]
ys = grid.cell_coords[..., 1]
np.testing.assert_allclose(tf.data[0, 1], 1)
np.testing.assert_allclose(tf.data[0, 1], 1)
np.testing.assert_allclose(tf.data[1, 0], xs**2)
np.testing.assert_allclose(tf.data[1, 1], xs * ys)
# corner case
with pytest.raises(ValueError):
Tensor2Field.from_expression(grid, "xy")
with pytest.raises(ValueError):
Tensor2Field.from_expression(grid, ["xy"])
with pytest.raises(ValueError):
Tensor2Field.from_expression(grid, ["x"] * 3)
with pytest.raises(ValueError):
Tensor2Field.from_expression(grid, [["x"], [1, 1]])
def test_examples_vector_sph():
"""compare derivatives of vector fields for spherical grids"""
grid = SphericalSymGrid(1, 32)
# divergence
vf = VectorField.from_expression(grid, ["r**3", 0, "r**2"])
res = vf.divergence([{"derivative": 0}, {"value": 1}])
expect = ScalarField.from_expression(grid, "5 * r**2")
np.testing.assert_allclose(res.data, expect.data, rtol=0.1, atol=0.1)
# vector gradient
vf = VectorField.from_expression(grid, ["r**3", 0, 0])
res = vf.gradient([{"derivative": 0}, {"value": [1, 1, 1]}])
expr = [["3 * r**2", 0, 0], [0, "r**2", 0], [0, 0, "r**2"]]
expect = Tensor2Field.from_expression(grid, expr)
np.testing.assert_allclose(res.data, expect.data, rtol=0.1, atol=0.1)
def test_examples_tensor_sph(conservative):
"""compare derivatives of tensorial fields for spherical grids"""
# test explicit expression for which we know the results
grid = SphericalSymGrid(1, 32)
expressions = [["r**4", 0, 0], [0, "r**3", 0], [0, 0, "r**3"]]
tf = Tensor2Field.from_expression(grid, expressions)
# tensor divergence
bc = [{"derivative": 0}, {"normal_derivative": [4, 3, 3]}]
res = tf.divergence(bc, conservative=conservative)
expect = VectorField.from_expression(grid,
["2 * r**2 * (3 * r - 1)", 0, 0])
if conservative:
np.testing.assert_allclose(res.data, expect.data, rtol=0.1, atol=0.1)
else:
np.testing.assert_allclose(res.data[:, 1:-1],
expect.data[:, 1:-1],
rtol=0.1,
atol=0.1)
def test_examples_tensor_cyl():
"""compare derivatives of tensorial fields for cylindrical grids"""
grid = CylindricalSymGrid(1, [0, 2 * np.pi], 32, periodic_z=True)
tf = Tensor2Field.from_expression(grid, [["r**3 * sin(z)"] * 3] * 3)
# tensor divergence
rs, zs = grid.axes_coords
val_r_outer = np.broadcast_to(6 * rs * np.sin(zs), (3, 32))
bcs = [({"derivative": 0}, {"curvature": val_r_outer}), "periodic"]
res = tf.divergence(bcs)
expect = VectorField.from_expression(
grid,
[
"r**2 * (r * cos(z) + 3 * sin(z))",
"r**2 * (r * cos(z) + 4 * sin(z))",
"r**2 * (r * cos(z) + 5 * sin(z))",
],
)
np.testing.assert_allclose(res.data, expect.data, rtol=0.1, atol=0.1)
def test_examples_vector_polar():
"""compare derivatives of vector fields for polar grids"""
grid = PolarSymGrid(1, 32)
vf = VectorField.from_expression(grid, ["r**3", "r**2"])
# divergence
res = vf.divergence([{"derivative": 0}, {"value": 1}])
expect = ScalarField.from_expression(grid, "4 * r**2")
np.testing.assert_allclose(res.data, expect.data, rtol=0.1, atol=0.1)
# # vector Laplacian
# res = vf.laplace([{"derivative": 0}, {"value": 1}])
# expect = VectorField.from_expression(grid, ["8 * r", "3"])
# np.testing.assert_allclose(res.data, expect.data, rtol=0.1, atol=0.1)
# vector gradient
res = vf.gradient([{"derivative": 0}, {"value": [1, 1]}])
expr = [["3 * r**2", "-r"], ["2 * r", "r**2"]]
expect = Tensor2Field.from_expression(grid, expr)
np.testing.assert_allclose(res.data, expect.data, rtol=0.1, atol=0.1)
def test_examples_vector_cyl():
"""compare derivatives of vector fields for cylindrical grids"""
grid = CylindricalSymGrid(1, [0, 2 * np.pi], 32)
e_r = "r**3 * sin(z)"
e_φ = "r**2 * sin(z)"
e_z = "r**4 * cos(z)"
vf = VectorField.from_expression(grid, [e_r, e_z, e_φ])
bc_r = ({"derivative_normal": 0}, {"value_normal": "r**3 * sin(z)"})
bc_z = {"curvature_normal": "-r**4 * cos(z)"}
bcs = [bc_r, bc_z]
# divergence
res = vf.divergence(bcs)
expect = ScalarField.from_expression(grid,
"4 * r**2 * sin(z) - r**4 * sin(z)")
np.testing.assert_allclose(res.data, expect.data, rtol=0.1, atol=0.1)
# vector Laplacian
grid = CylindricalSymGrid(1, [0, 2 * np.pi], 32, periodic_z=True)
vf = VectorField.from_expression(grid, ["r**3 * sin(z)"] * 3)
val_r_outer = np.broadcast_to(6 * np.sin(grid.axes_coords[1]), (3, 32))
bcs = [({"derivative": 0}, {"curvature": val_r_outer}), "periodic"]
res = vf.laplace(bcs)
expr = [
"8 * r * sin(z) - r**3 * sin(z)",
"9 * r * sin(z) - r**3 * sin(z)",
"8 * r * sin(z) - r**3 * sin(z)",
]
expect = VectorField.from_expression(grid, expr)
np.testing.assert_allclose(res.data, expect.data, rtol=0.1, atol=0.1)
# vector gradient
bcs = [({"derivative": 0}, {"curvature": val_r_outer}), "periodic"]
res = vf.gradient(bcs)
expr = [
["3 * r**2 * sin(z)", "r**3 * cos(z)", "-r**2 * sin(z)"],
["3 * r**2 * sin(z)", "r**3 * cos(z)", 0],
["3 * r**2 * sin(z)", "r**3 * cos(z)", "r**2 * sin(z)"],
]
expect = Tensor2Field.from_expression(grid, expr)
np.testing.assert_allclose(res.data, expect.data, rtol=0.1, atol=0.1)
def test_conservative_sph():
"""test whether the integral over a divergence vanishes"""
grid = SphericalSymGrid((0, 2), 50)
expr = "1 / cosh((r - 1) * 10)"
# test divergence of vector field
vf = VectorField.from_expression(grid, [expr, 0, 0])
div = vf.divergence(bc="derivative", conservative=True)
assert div.integral == pytest.approx(0, abs=1e-2)
# test laplacian of scalar field
lap = vf[0].laplace("derivative")
assert lap.integral == pytest.approx(0, abs=1e-13)
# test double divergence of tensor field
expressions = [[expr, 0, 0], [0, expr, 0], [0, 0, expr]]
tf = Tensor2Field.from_expression(grid, expressions)
res = tf._apply_operator("tensor_double_divergence",
bc="derivative",
conservative=True)
assert res.integral == pytest.approx(0, abs=1e-3)