def test_gradient(self):
mesh = get_unit_2p1D_mesh_for_testing(ypts=32, zpts=32, include_particles=False)
spatial_methods = {
"macroscale": pybamm.FiniteVolume(),
"current collector": pybamm.ScikitFiniteElement(),
}
disc = pybamm.Discretisation(mesh, spatial_methods)
# test gradient of 5*y + 6*z
var = pybamm.Variable("var", domain="current collector")
disc.set_variable_slices([var])
y = mesh["current collector"].coordinates[0, :]
z = mesh["current collector"].coordinates[1, :]
gradient = pybamm.grad(var)
grad_disc = disc.process_symbol(gradient)
grad_disc_y, grad_disc_z = grad_disc.children
np.testing.assert_array_almost_equal(
grad_disc_y.evaluate(None, 5 * y + 6 * z),
5 * np.ones_like(y)[:, np.newaxis],
)
np.testing.assert_array_almost_equal(
grad_disc_z.evaluate(None, 5 * y + 6 * z),
6 * np.ones_like(z)[:, np.newaxis],
)
# check grad_squared positive
eqn = pybamm.grad_squared(var)
eqn_disc = disc.process_symbol(eqn)
ans = eqn_disc.evaluate(None, 3 * y ** 2)
np.testing.assert_array_less(0, ans)
def test_disc_spatial_var(self):
mesh = get_unit_2p1D_mesh_for_testing(ypts=4,
zpts=5,
include_particles=False)
spatial_methods = {
"macroscale": pybamm.FiniteVolume(),
"current collector": pybamm.ScikitFiniteElement(),
}
disc = pybamm.Discretisation(mesh, spatial_methods)
# discretise y and z
y = pybamm.SpatialVariable("y", ["current collector"])
z = pybamm.SpatialVariable("z", ["current collector"])
y_disc = disc.process_symbol(y)
z_disc = disc.process_symbol(z)
# create expected meshgrid
y_vec = np.linspace(0, 1, 4)
z_vec = np.linspace(0, 1, 5)
Y, Z = np.meshgrid(y_vec, z_vec)
y_actual = np.transpose(Y).flatten()[:, np.newaxis]
z_actual = np.transpose(Z).flatten()[:, np.newaxis]
# spatial vars should discretise to the flattend meshgrid
np.testing.assert_array_equal(y_disc.evaluate(), y_actual)
np.testing.assert_array_equal(z_disc.evaluate(), z_actual)
def test_pure_neumann_poisson(self):
# grad^2 u = 1, du/dz = 1 at z = 1, du/dn = 0 elsewhere, u has zero average
u = pybamm.Variable("u", domain="current collector")
c = pybamm.Variable("c") # lagrange multiplier
y = pybamm.SpatialVariable("y", ["current collector"])
z = pybamm.SpatialVariable("z", ["current collector"])
model = pybamm.BaseModel()
# 0*c hack otherwise gives KeyError
model.algebraic = {
u:
pybamm.laplacian(u) - pybamm.source(1, u) +
c * pybamm.DefiniteIntegralVector(u, vector_type="column"),
c:
pybamm.Integral(u, [y, z]) + 0 * c,
}
model.initial_conditions = {u: pybamm.Scalar(0), c: pybamm.Scalar(0)}
# set boundary conditions ("negative tab" = bottom of unit square,
# "positive tab" = top of unit square, elsewhere normal derivative is zero)
model.boundary_conditions = {
u: {
"negative tab": (0, "Neumann"),
"positive tab": (1, "Neumann")
}
}
model.variables = {"c": c, "u": u}
# create discretisation
mesh = get_unit_2p1D_mesh_for_testing(ypts=32,
zpts=32,
include_particles=False)
spatial_methods = {
"macroscale": pybamm.FiniteVolume(),
"current collector": pybamm.ScikitFiniteElement(),
}
disc = pybamm.Discretisation(mesh, spatial_methods)
disc.process_model(model)
# solve model
solver = pybamm.AlgebraicSolver()
solution = solver.solve(model)
z = mesh["current collector"].coordinates[1, :][:, np.newaxis]
u_exact = z**2 / 2 - 1 / 6
np.testing.assert_array_almost_equal(solution.y[:-1],
u_exact,
decimal=1)
def test_dirichlet_bcs(self):
# manufactured solution u = a*z^2 + b*z + c
model = pybamm.BaseModel()
a = 3
b = 4
c = 5
u = pybamm.Variable("variable", domain="current collector")
model.algebraic = {u: -pybamm.laplacian(u) + pybamm.source(2 * a, u)}
# set boundary conditions ("negative tab" = bottom of unit square,
# "positive tab" = top of unit square, elsewhere normal derivative is zero)
model.boundary_conditions = {
u: {
"negative tab": (pybamm.Scalar(c), "Dirichlet"),
"positive tab": (pybamm.Scalar(a + b + c), "Dirichlet"),
}
}
# bad initial guess (on purpose)
model.initial_conditions = {u: pybamm.Scalar(1)}
model.variables = {"u": u}
# create discretisation
mesh = get_unit_2p1D_mesh_for_testing(ypts=8,
zpts=32,
include_particles=False)
spatial_methods = {
"macroscale": pybamm.FiniteVolume(),
"current collector": pybamm.ScikitFiniteElement(),
}
disc = pybamm.Discretisation(mesh, spatial_methods)
disc.process_model(model)
# solve model
solver = pybamm.AlgebraicSolver()
solution = solver.solve(model)
# indepedent of y, so just check values for one y
z = mesh["current collector"].edges["z"][:, np.newaxis]
u_exact = a * z**2 + b * z + c
np.testing.assert_array_almost_equal(solution.y[0:len(z)], u_exact)
def test_manufactured_solution(self):
mesh = get_unit_2p1D_mesh_for_testing(ypts=32, zpts=32, include_particles=False)
spatial_methods = {
"macroscale": pybamm.FiniteVolume(),
"current collector": pybamm.ScikitFiniteElement(),
}
disc = pybamm.Discretisation(mesh, spatial_methods)
# linear u = z (to test coordinates to degree of freedom mapping)
var = pybamm.Variable("var", domain="current collector")
disc.set_variable_slices([var])
var_disc = disc.process_symbol(var)
z_vertices = mesh["current collector"].coordinates[1, :]
np.testing.assert_array_almost_equal(
var_disc.evaluate(None, z_vertices), z_vertices[:, np.newaxis]
)
# linear u = 6*y (to test coordinates to degree of freedom mapping)
y_vertices = mesh["current collector"].coordinates[0, :]
np.testing.assert_array_almost_equal(
var_disc.evaluate(None, 6 * y_vertices), 6 * y_vertices[:, np.newaxis]
)
# mixed u = y*z (to test coordinates to degree of freedom mapping)
np.testing.assert_array_almost_equal(
var_disc.evaluate(None, y_vertices * z_vertices),
y_vertices[:, np.newaxis] * z_vertices[:, np.newaxis],
)
# laplace of u = sin(pi*z)
var = pybamm.Variable("var", domain="current collector")
eqn_zz = pybamm.laplacian(var)
# set boundary conditions ("negative tab" = bottom of unit square,
# "positive tab" = top of unit square, elsewhere normal derivative is zero)
disc.bcs = {
var.id: {
"negative tab": (pybamm.Scalar(0), "Dirichlet"),
"positive tab": (pybamm.Scalar(0), "Dirichlet"),
}
}
disc.set_variable_slices([var])
eqn_zz_disc = disc.process_symbol(eqn_zz)
z_vertices = mesh["current collector"].coordinates[1, :][:, np.newaxis]
u = np.sin(np.pi * z_vertices)
mass = pybamm.Mass(var)
mass_disc = disc.process_symbol(mass)
soln = -np.pi ** 2 * u
np.testing.assert_array_almost_equal(
eqn_zz_disc.evaluate(None, u), mass_disc.entries @ soln, decimal=3
)
# laplace of u = cos(pi*y)*sin(pi*z)
var = pybamm.Variable("var", domain="current collector")
laplace_eqn = pybamm.laplacian(var)
# set boundary conditions ("negative tab" = bottom of unit square,
# "positive tab" = top of unit square, elsewhere normal derivative is zero)
disc.bcs = {
var.id: {
"negative tab": (pybamm.Scalar(0), "Dirichlet"),
"positive tab": (pybamm.Scalar(0), "Dirichlet"),
}
}
disc.set_variable_slices([var])
laplace_eqn_disc = disc.process_symbol(laplace_eqn)
y_vertices = mesh["current collector"].coordinates[0, :][:, np.newaxis]
z_vertices = mesh["current collector"].coordinates[1, :][:, np.newaxis]
u = np.cos(np.pi * y_vertices) * np.sin(np.pi * z_vertices)
mass = pybamm.Mass(var)
mass_disc = disc.process_symbol(mass)
soln = -np.pi ** 2 * u
np.testing.assert_array_almost_equal(
laplace_eqn_disc.evaluate(None, u), mass_disc.entries @ soln, decimal=2
)
