def set_algebraic(self, variables):
param = self.param
applied_current = variables["Total current density"]
cc_area = self._get_effective_current_collector_area()
z = pybamm.standard_spatial_vars.z
phi_s_cn = variables["Negative current collector potential"]
phi_s_cp = variables["Positive current collector potential"]
i_boundary_cc = variables["Current collector current density"]
i_boundary_cc_0 = variables[
"Leading-order current collector current density"]
c = variables["Lagrange multiplier"]
# Note that the second argument of 'source' must be the same as the argument
# in the laplacian (the variable to which the boundary conditions are applied)
self.algebraic = {
phi_s_cn: (param.sigma_cn * param.delta**2 * param.l_cn) *
pybamm.laplacian(phi_s_cn) -
pybamm.source(i_boundary_cc_0, phi_s_cn),
i_boundary_cc: (param.sigma_cp * param.delta**2 * param.l_cp) *
pybamm.laplacian(phi_s_cp) +
pybamm.source(i_boundary_cc_0, phi_s_cp) +
c * pybamm.PrimaryBroadcast(cc_area, "current collector"),
c:
pybamm.Integral(i_boundary_cc, z) - applied_current / cc_area +
0 * c,
}
def set_rhs(self, variables):
T_av = variables["X-averaged cell temperature"]
Q_av = variables["X-averaged total heating"]
T_amb = variables["Ambient temperature"]
# Account for surface area to volume ratio of pouch cell in cooling
# coefficient. Note: the factor 1/delta^2 comes from the choice of
# non-dimensionalisation
yz_surface_area = self.param.l_y * self.param.l_z
cell_volume = self.param.l * self.param.l_y * self.param.l_z
yz_surface_cooling_coefficient = (
-(self.param.h_cn + self.param.h_cp) * yz_surface_area /
cell_volume / (self.param.delta**2))
edge_cooling_coefficient = self.param.h_edge / self.param.delta
# Governing equations contain:
# - source term for y-z surface cooling
# - boundary source term of edge cooling
# Boundary conditions contain:
# - Neumann condition for tab cooling
self.rhs = {
T_av:
(pybamm.laplacian(T_av) + self.param.B * pybamm.source(Q_av, T_av)
+ yz_surface_cooling_coefficient *
pybamm.source(T_av - T_amb, T_av) - edge_cooling_coefficient *
pybamm.source(T_av - T_amb, T_av, boundary=True)) /
(self.param.C_th * self.param.rho)
}
def set_algebraic(self, variables):
R_cn_scaled = variables["Scaled negative current collector resistance"]
R_cp_scaled = variables["Scaled positive current collector resistance"]
self.algebraic = {
R_cn_scaled: pybamm.laplacian(R_cn_scaled) - pybamm.source(1, R_cn_scaled),
R_cp_scaled: pybamm.laplacian(R_cp_scaled) - pybamm.source(1, R_cp_scaled),
}
def test_source(self):
u = pybamm.Variable("u", domain="current collector")
v = pybamm.Variable("v", domain="current collector")
source = pybamm.source(u, v)
self.assertIsInstance(source.children[0], pybamm.Mass)
boundary_source = pybamm.source(u, v, boundary=True)
self.assertIsInstance(boundary_source.children[0], pybamm.BoundaryMass)
def set_rhs(self, variables):
T_av = variables["X-averaged cell temperature"]
Q_av = variables["X-averaged total heating"]
# Add boundary source term which accounts for surface cooling around
# the edge of the domain in the weak formulation.
# TODO: update to allow different cooling conditions at the tabs
self.rhs = {
T_av:
(pybamm.laplacian(T_av) +
self.param.B * pybamm.source(Q_av, T_av) -
(2 * self.param.h / (self.param.delta**2) / self.param.l) *
pybamm.source(T_av, T_av) - (self.param.h / self.param.delta) *
pybamm.source(T_av, T_av, boundary=True)) / self.param.C_th
}
def set_algebraic(self, variables):
param = self.param
phi_s_cn = variables["Negative current collector potential"]
phi_s_cp = variables["Positive current collector potential"]
i_boundary_cc = variables["Current collector current density"]
self.algebraic = {
phi_s_cn: (param.sigma_cn * param.delta ** 2 * param.l_cn)
* pybamm.laplacian(phi_s_cn)
- pybamm.source(i_boundary_cc, phi_s_cn),
i_boundary_cc: (param.sigma_cp * param.delta ** 2 * param.l_cp)
* pybamm.laplacian(phi_s_cp)
+ pybamm.source(i_boundary_cc, phi_s_cp),
}
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_source_error(self):
# test error with domain not current collector
v = pybamm.Vector(np.ones(5), domain="current collector")
w = pybamm.Vector(2 * np.ones(3), domain="test")
with self.assertRaisesRegex(pybamm.DomainError, "'source'"):
pybamm.source(v, w)
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")
def test_discretise_equations(self):
# get mesh
mesh = get_2p1d_mesh_for_testing(include_particles=False)
spatial_methods = {
"macroscale": pybamm.FiniteVolume(),
"current collector": pybamm.ScikitFiniteElement(),
}
disc = pybamm.Discretisation(mesh, spatial_methods)
# discretise some equations
var = pybamm.Variable("var", domain="current collector")
y = pybamm.SpatialVariable("y", ["current collector"])
z = pybamm.SpatialVariable("z", ["current collector"])
disc.set_variable_slices([var])
y_test = np.ones(mesh["current collector"].npts)
unit_source = pybamm.PrimaryBroadcast(1, "current collector")
disc.bcs = {
var.id: {
"negative tab": (pybamm.Scalar(0), "Neumann"),
"positive tab": (pybamm.Scalar(0), "Neumann"),
}
}
for eqn in [
pybamm.laplacian(var),
pybamm.source(unit_source, var),
pybamm.laplacian(var) - pybamm.source(unit_source, var),
pybamm.source(var, var),
pybamm.laplacian(var) - pybamm.source(2 * var, var),
pybamm.laplacian(var) - pybamm.source(unit_source ** 2 + 1 / var, var),
pybamm.Integral(var, [y, z]) - 1,
pybamm.source(var, var, boundary=True),
pybamm.laplacian(var) - pybamm.source(unit_source, var, boundary=True),
pybamm.laplacian(var)
- pybamm.source(unit_source ** 2 + 1 / var, var, boundary=True),
]:
# Check that equation can be evaluated in each case
# Dirichlet
disc.bcs = {
var.id: {
"negative tab": (pybamm.Scalar(0), "Dirichlet"),
"positive tab": (pybamm.Scalar(1), "Dirichlet"),
}
}
eqn_disc = disc.process_symbol(eqn)
eqn_disc.evaluate(None, y_test)
# Neumann
disc.bcs = {
var.id: {
"negative tab": (pybamm.Scalar(0), "Neumann"),
"positive tab": (pybamm.Scalar(1), "Neumann"),
}
}
eqn_disc = disc.process_symbol(eqn)
eqn_disc.evaluate(None, y_test)
# One of each
disc.bcs = {
var.id: {
"negative tab": (pybamm.Scalar(0), "Neumann"),
"positive tab": (pybamm.Scalar(1), "Dirichlet"),
}
}
eqn_disc = disc.process_symbol(eqn)
eqn_disc.evaluate(None, y_test)
# One of each
disc.bcs = {
var.id: {
"negative tab": (pybamm.Scalar(0), "Dirichlet"),
"positive tab": (pybamm.Scalar(1), "Neumann"),
}
}
eqn_disc = disc.process_symbol(eqn)
eqn_disc.evaluate(None, y_test)
# check ValueError raised for non Dirichlet or Neumann BCs
eqn = pybamm.laplacian(var) - pybamm.source(unit_source, var)
disc.bcs = {
var.id: {
"negative tab": (pybamm.Scalar(0), "Dirichlet"),
"positive tab": (pybamm.Scalar(1), "Other BC"),
}
}
with self.assertRaises(ValueError):
eqn_disc = disc.process_symbol(eqn)
disc.bcs = {
var.id: {
"negative tab": (pybamm.Scalar(0), "Other BC"),
"positive tab": (pybamm.Scalar(1), "Neumann"),
}
}
with self.assertRaises(ValueError):
eqn_disc = disc.process_symbol(eqn)
# raise ModelError if no BCs provided
new_var = pybamm.Variable("new_var", domain="current collector")
disc.set_variable_slices([new_var])
eqn = pybamm.laplacian(new_var)
with self.assertRaises(pybamm.ModelError):
eqn_disc = disc.process_symbol(eqn)
# check GeometryError if using scikit-fem not in y or z
x = pybamm.SpatialVariable("x", ["current collector"])
with self.assertRaises(pybamm.GeometryError):
disc.process_symbol(x)
def set_rhs(self, variables):
# Note: we have to use `pybamm.source(rhs, var)` in the rhs dict so that
# the scalar source term gets multplied by the correct mass matrix when
# using this model with 2D current collectors with the finite element
# method (see #1399)
c_s_rxav = variables[
"Average " + self.domain.lower() + " particle concentration"
]
j_xav = variables[
"X-averaged "
+ self.domain.lower()
+ " electrode interfacial current density"
]
if self.domain == "Negative":
self.rhs = {
c_s_rxav: pybamm.source(-3 * j_xav / self.param.a_R_n, c_s_rxav)
}
elif self.domain == "Positive":
self.rhs = {
c_s_rxav: pybamm.source(
-3 * j_xav / self.param.a_R_p / self.param.gamma_p, c_s_rxav
)
}
if self.name == "quartic profile":
# We solve an extra ODE for the average particle concentration gradient
q_s_rxav = variables[
"Average " + self.domain.lower() + " particle concentration gradient"
]
c_s_surf_xav = variables[
"X-averaged " + self.domain.lower() + " particle surface concentration"
]
T_xav = variables[
"X-averaged " + self.domain.lower() + " electrode temperature"
]
if self.domain == "Negative":
self.rhs.update(
{
q_s_rxav: pybamm.source(
-30
* self.param.D_n(c_s_surf_xav, T_xav)
* q_s_rxav
/ self.param.C_n
- 45 * j_xav / self.param.a_R_n / 2,
q_s_rxav,
)
}
)
elif self.domain == "Positive":
self.rhs.update(
{
q_s_rxav: pybamm.source(
-30
* self.param.D_p(c_s_surf_xav, T_xav)
* q_s_rxav
/ self.param.C_p
- 45 * j_xav / self.param.a_R_p / self.param.gamma_p / 2,
q_s_rxav,
)
}
)
