def test_definite_integral_vector(self):
mesh = get_2p1d_mesh_for_testing(include_particles=False)
spatial_methods = {
"macroscale": pybamm.FiniteVolume(),
"current collector": pybamm.ScikitFiniteElement(),
}
disc = pybamm.Discretisation(mesh, spatial_methods)
var = pybamm.Variable("var", domain="current collector")
disc.set_variable_slices([var])
# row (default)
vec = pybamm.DefiniteIntegralVector(var)
vec_disc = disc.process_symbol(vec)
self.assertEqual(vec_disc.shape[0], 1)
self.assertEqual(vec_disc.shape[1], mesh["current collector"].npts)
# column
vec = pybamm.DefiniteIntegralVector(var, vector_type="column")
vec_disc = disc.process_symbol(vec)
self.assertEqual(vec_disc.shape[0], mesh["current collector"].npts)
self.assertEqual(vec_disc.shape[1], 1)
def test_definite_integral_vector(self):
mesh = get_mesh_for_testing()
spatial_methods = {
"macroscale": pybamm.FiniteVolume(),
"negative particle": pybamm.FiniteVolume(),
"positive particle": pybamm.FiniteVolume(),
}
disc = pybamm.Discretisation(mesh, spatial_methods)
var = pybamm.Variable("var", domain="negative electrode")
disc.set_variable_slices([var])
# row (default)
vec = pybamm.DefiniteIntegralVector(var)
vec_disc = disc.process_symbol(vec)
self.assertEqual(vec_disc.shape[0], 1)
self.assertEqual(vec_disc.shape[1], mesh["negative electrode"][0].npts)
# column
vec = pybamm.DefiniteIntegralVector(var, vector_type="column")
vec_disc = disc.process_symbol(vec)
self.assertEqual(vec_disc.shape[0], mesh["negative electrode"][0].npts)
self.assertEqual(vec_disc.shape[1], 1)
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 __init__(self):
super().__init__()
self.name = "Effective resistance in current collector model"
self.param = pybamm.standard_parameters_lithium_ion
# Get useful parameters
param = self.param
l_cn = param.l_cn
l_cp = param.l_cp
l_y = param.l_y
sigma_cn_dbl_prime = param.sigma_cn_dbl_prime
sigma_cp_dbl_prime = param.sigma_cp_dbl_prime
alpha_prime = param.alpha_prime
# Set model variables
var = pybamm.standard_spatial_vars
psi = pybamm.Variable("Current collector potential weighted sum",
["current collector"])
W = pybamm.Variable(
"Perturbation to current collector potential difference",
["current collector"],
)
c_psi = pybamm.Variable("Lagrange multiplier for variable `psi`")
c_W = pybamm.Variable("Lagrange multiplier for variable `W`")
self.variables = {
"Current collector potential weighted sum": psi,
"Perturbation to current collector potential difference": W,
"Lagrange multiplier for variable `psi`": c_psi,
"Lagrange multiplier for variable `W`": c_W,
}
# Algebraic equations (enforce zero mean constraint through Lagrange multiplier)
# 0*LagrangeMultiplier hack otherwise gives KeyError
self.algebraic = {
psi:
pybamm.laplacian(psi) +
c_psi * pybamm.DefiniteIntegralVector(psi, vector_type="column"),
W:
pybamm.laplacian(W) - pybamm.source(1, W) +
c_W * pybamm.DefiniteIntegralVector(W, vector_type="column"),
c_psi:
pybamm.Integral(psi, [var.y, var.z]) + 0 * c_psi,
c_W:
pybamm.Integral(W, [var.y, var.z]) + 0 * c_W,
}
# Boundary conditons
psi_neg_tab_bc = l_cn
psi_pos_tab_bc = -l_cp
W_neg_tab_bc = l_y / (alpha_prime * sigma_cn_dbl_prime)
W_pos_tab_bc = l_y / (alpha_prime * sigma_cp_dbl_prime)
self.boundary_conditions = {
psi: {
"negative tab": (psi_neg_tab_bc, "Neumann"),
"positive tab": (psi_pos_tab_bc, "Neumann"),
},
W: {
"negative tab": (W_neg_tab_bc, "Neumann"),
"positive tab": (W_pos_tab_bc, "Neumann"),
},
}
# "Initial conditions" provides initial guess for solver
# TODO: better guess than zero?
self.initial_conditions = {
psi: pybamm.Scalar(0),
W: pybamm.Scalar(0),
c_psi: pybamm.Scalar(0),
c_W: pybamm.Scalar(0),
}
# Define effective current collector resistance
psi_neg_tab = pybamm.BoundaryValue(psi, "negative tab")
psi_pos_tab = pybamm.BoundaryValue(psi, "positive tab")
W_neg_tab = pybamm.BoundaryValue(W, "negative tab")
W_pos_tab = pybamm.BoundaryValue(W, "positive tab")
R_cc = ((alpha_prime / l_y) * (sigma_cn_dbl_prime * l_cn * W_pos_tab +
sigma_cp_dbl_prime * l_cp * W_neg_tab) -
(psi_pos_tab - psi_neg_tab)) / (sigma_cn_dbl_prime * l_cn +
sigma_cp_dbl_prime * l_cp)
R_cc_dim = R_cc * param.potential_scale / param.I_typ
self.variables.update({
"Current collector potential weighted sum (negative tab)":
psi_neg_tab,
"Current collector potential weighted sum (positive tab)":
psi_pos_tab,
"Perturbation to c.c. potential difference (negative tab)":
W_neg_tab,
"Perturbation to c.c. potential difference (positive tab)":
W_pos_tab,
"Effective current collector resistance":
R_cc,
"Effective current collector resistance [Ohm]":
R_cc_dim,
})
def __init__(self):
super().__init__()
self.name = "Effective resistance in current collector model (2D)"
self.param = pybamm.standard_parameters_lithium_ion
# Get necessary parameters
param = self.param
l_cn = param.l_cn
l_cp = param.l_cp
l_tab_p = param.l_tab_p
A_tab_p = l_cp * l_tab_p
sigma_cn_dbl_prime = param.sigma_cn_dbl_prime
sigma_cp_dbl_prime = param.sigma_cp_dbl_prime
delta = param.delta
# Set model variables -- we solve a auxilliary problem in each current collector
# then relate this to the potentials and resistances later
f_n = pybamm.Variable("Unit solution in negative current collector",
domain="current collector")
f_p = pybamm.Variable("Unit solution in positive current collector",
domain="current collector")
# Governing equations -- we impose that the average of f_p is zero
# by introducing a Lagrange multiplier
c = pybamm.Variable("Lagrange multiplier")
self.algebraic = {
f_n:
pybamm.laplacian(f_n) + pybamm.source(1, f_n),
c:
pybamm.laplacian(f_p) - pybamm.source(1, f_p) +
c * pybamm.DefiniteIntegralVector(f_p, vector_type="column"),
f_p:
pybamm.yz_average(f_p) + 0 * c,
}
# Boundary conditons
pos_tab_bc = l_cp / A_tab_p
self.boundary_conditions = {
f_n: {
"negative tab": (0, "Dirichlet"),
"positive tab": (0, "Neumann")
},
f_p: {
"negative tab": (0, "Neumann"),
"positive tab": (pos_tab_bc, "Neumann"),
},
}
# "Initial conditions" provides initial guess for solver
self.initial_conditions = {
f_n: pybamm.Scalar(0),
f_p: pybamm.Scalar(0),
c: pybamm.Scalar(0),
}
# Define effective current collector resistance
R_cc_n = delta * pybamm.yz_average(f_n) / (l_cn * sigma_cn_dbl_prime)
R_cc_p = (delta * pybamm.BoundaryIntegral(f_p, "positive tab") /
(l_cp * sigma_cp_dbl_prime))
R_cc = R_cc_n + R_cc_p
R_scale = param.potential_scale / param.I_typ
self.variables = {
"Unit solution in negative current collector":
f_n,
"Unit solution in positive current collector":
f_p,
"Effective current collector resistance":
R_cc,
"Effective current collector resistance [Ohm]":
R_cc * R_scale,
"Effective negative current collector resistance":
R_cc_n,
"Effective negative current collector resistance [Ohm]":
R_cc_n * R_scale,
"Effective positive current collector resistance":
R_cc_p,
"Effective positive current collector resistance [Ohm]":
R_cc_p * R_scale,
}
# Add spatial variables
var = pybamm.standard_spatial_vars
L_y = pybamm.geometric_parameters.L_y
L_z = pybamm.geometric_parameters.L_z
self.variables.update({
"y": var.y,
"y [m]": var.y * L_y,
"z": var.z,
"z [m]": var.z * L_z
})
pybamm.citations.register("timms2020")
