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": 0}, {"normal_value": [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)
res = tf.divergence([{"derivative": 0}, {"value": np.ones((2, 2))}])
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_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_scalar_polar():
"""compare derivatives of scalar fields for polar grids"""
grid = PolarSymGrid(1, 32)
sf = ScalarField.from_expression(grid, "r**3")
# gradient
res = sf.gradient([{"derivative": 0}, {"derivative": 3}])
expect = VectorField.from_expression(grid, ["3 * r**2", 0])
np.testing.assert_allclose(res.data, expect.data, rtol=0.1, atol=0.1)
# gradient squared
expect = ScalarField.from_expression(grid, "9 * r**4")
for c in [True, False]:
res = sf.gradient_squared([{
"derivative": 0
}, {
"derivative": 3
}],
central=c)
np.testing.assert_allclose(res.data, expect.data, rtol=0.1, atol=0.1)
# laplace
res = sf.laplace([{"derivative": 0}, {"derivative": 3}])
expect = ScalarField.from_expression(grid, "9 * r")
np.testing.assert_allclose(res.data, expect.data, rtol=0.1, atol=0.1)
def test_examples_scalar_cyl():
"""compare derivatives of scalar fields for cylindrical grids"""
grid = CylindricalSymGrid(1, [0, 2 * np.pi], 32)
expr = "r**3 * sin(z)"
sf = ScalarField.from_expression(grid, expr)
bcs = [[{
"derivative": 0
}, {
"value": expr
}], [{
"value": expr
}, {
"value": expr
}]]
# gradient - The coordinates are ordered as (r, z, φ) in py-pde
res = sf.gradient(bcs)
expect = VectorField.from_expression(
grid, ["3 * r**2 * sin(z)", "r**3 * cos(z)", 0])
np.testing.assert_allclose(res.data, expect.data, rtol=0.1, atol=0.1)
# gradient squared
expect = ScalarField.from_expression(
grid, "r**6 * cos(z)**2 + 9 * r**4 * sin(z)**2")
res = sf.gradient_squared(bcs, central=True)
np.testing.assert_allclose(res.data, expect.data, rtol=0.1, atol=0.1)
# laplace
bcs[0][1] = {
"curvature": "6 * sin(z)"
} # adjust BC to fit laplacian better
res = sf.laplace(bcs)
expect = ScalarField.from_expression(grid,
"9 * r * sin(z) - r**3 * sin(z)")
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_tensor_sph_symmetry():
"""test treatment of symmetric tensor field"""
grid = SphericalSymGrid(1, 16)
vf = VectorField.from_expression(grid, ["r**2", 0, 0])
vf_grad = vf.gradient({"derivative": 2})
strain = vf_grad + vf_grad.transpose()
bcs = [{"value": 0}, {"normal_derivative": [4, 0, 0]}]
strain_div = strain.divergence(bcs, safe=False)
np.testing.assert_allclose(strain_div.data[0], 8)
np.testing.assert_allclose(strain_div.data[1:], 0)
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_divergence_field_cyl():
"""test the divergence operator"""
grid = CylindricalSymGrid(2 * np.pi, [0, 2 * np.pi], [16, 32], periodic_z=True)
v = VectorField.from_expression(grid, ["cos(r) + sin(z)**2", "z * cos(r)**2", 0])
s = v.divergence(bc="natural")
assert s.data.shape == grid.shape
res = ScalarField.from_expression(
grid, "cos(r)**2 - sin(r) + (cos(r) + sin(z)**2) / r"
)
np.testing.assert_allclose(
s.data[1:-1, 1:-1], res.data[1:-1, 1:-1], rtol=0.1, atol=0.1
)
def test_vector_boundary_conditions():
"""test some boundary conditions of operators of vector fields"""
grid = CartesianGrid([[0, 2 * np.pi], [0, 1]], 32, periodic=[False, True])
vf = VectorField.from_expression(grid, ["sin(x)", "0"])
bc_x = [{"derivative": [-1, 0]}, {"derivative": [1, 0]}]
tf = vf.gradient(bc=[bc_x, "periodic"])
res = ScalarField.from_expression(grid, "cos(x)")
np.testing.assert_allclose(tf[0, 0].data, res.data, atol=0.01, rtol=0.01)
np.testing.assert_allclose(tf[0, 1].data, 0)
np.testing.assert_allclose(tf[1, 0].data, 0)
np.testing.assert_allclose(tf[1, 1].data, 0)
def test_vector_bcs():
"""test boundary conditions on vector fields"""
grid = UnitGrid([3, 3], periodic=False)
v = VectorField.from_expression(grid, ["x", "cos(y)"])
bc_x = {"value": [0, 1]}
bc_y = {"value": [2, 3]}
bcs = [bc_x, bc_y]
s1 = v.divergence(bcs, backend="scipy").data
div = grid.make_operator("divergence", bcs)
s2 = div(v.data)
np.testing.assert_allclose(s1, s2)
def test_evaluate_func_vector():
"""test the evaluate function with vector fields"""
grid = UnitGrid([3])
field = ScalarField.from_expression(grid, "x")
vec = VectorField.from_expression(grid, ["x"])
res = evaluate("inner(v, v)", {"v": vec})
assert isinstance(res, ScalarField)
np.testing.assert_almost_equal(res.data, grid.axes_coords[0] ** 2)
res = evaluate("outer(v, v)", {"v": vec})
assert isinstance(res, Tensor2Field)
np.testing.assert_almost_equal(res.data, [[grid.axes_coords[0] ** 2]])
assert isinstance(evaluate("gradient(a)", {"a": field}), VectorField)
def test_from_expressions():
"""test initializing vector fields with expressions"""
grid = UnitGrid([4, 4])
vf = VectorField.from_expression(grid, ["x**2", "x * y"])
xs = grid.cell_coords[..., 0]
ys = grid.cell_coords[..., 1]
np.testing.assert_allclose(vf.data[0], xs**2)
np.testing.assert_allclose(vf.data[1], xs * ys)
# corner case
vf = VectorField.from_expression(grid, ["1", "x * y"])
np.testing.assert_allclose(vf.data[0], 1)
vf = VectorField.from_expression(grid, [1, "x * y"])
np.testing.assert_allclose(vf.data[0], 1)
with pytest.raises(ValueError):
VectorField.from_expression(grid, "xy")
with pytest.raises(ValueError):
VectorField.from_expression(grid, ["xy"])
with pytest.raises(ValueError):
VectorField.from_expression(grid, ["x"] * 3)
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_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)
r""" Plotting a vector field ======================= This example shows how to initialize and visualize the vector field :math:`\boldsymbol u = \bigl(\sin(x), \cos(x)\bigr)`. """ from pde import CartesianGrid, VectorField grid = CartesianGrid([[-2, 2], [-2, 2]], 32) field = VectorField.from_expression(grid, ['sin(x)', 'cos(x)']) field.plot(method='streamplot', title='Stream plot')
r""" Plotting a vector field ======================= This example shows how to initialize and visualize the vector field :math:`\boldsymbol u = \bigl(\sin(x), \cos(x)\bigr)`. """ from pde import CartesianGrid, VectorField grid = CartesianGrid([[-2, 2], [-2, 2]], 32) field = VectorField.from_expression(grid, ["sin(x)", "cos(x)"]) field.plot(method="streamplot", title="Stream plot")
