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 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 set_soc_variables(self):
"Set variables relating to the state of charge."
# State of Charge defined as function of dimensionless electrolyte concentration
z = pybamm.standard_spatial_vars.z
soc = (pybamm.Integral(
self.variables["X-averaged electrolyte concentration"], z) * 100)
self.variables.update({
"State of Charge": soc,
"Depth of Discharge": 100 - soc
})
# Fractional charge input
if "Fractional Charge Input" not in self.variables:
fci = pybamm.Variable("Fractional Charge Input",
domain="current collector")
self.variables["Fractional Charge Input"] = fci
self.rhs[fci] = -self.variables["Total current density"] * 100
self.initial_conditions[fci] = self.param.q_init * 100
def test_delta_function(self):
mesh = get_mesh_for_testing()
spatial_methods = {"macroscale": pybamm.FiniteVolume()}
disc = pybamm.Discretisation(mesh, spatial_methods)
var = pybamm.Variable("var")
delta_fn_left = pybamm.DeltaFunction(var, "left", "negative electrode")
delta_fn_right = pybamm.DeltaFunction(var, "right",
"negative electrode")
disc.set_variable_slices([var])
delta_fn_left_disc = disc.process_symbol(delta_fn_left)
delta_fn_right_disc = disc.process_symbol(delta_fn_right)
# Basic shape and type tests
y = np.ones_like(mesh["negative electrode"][0].nodes[:, np.newaxis])
# Left
self.assertEqual(delta_fn_left_disc.domain, delta_fn_left.domain)
self.assertEqual(delta_fn_left_disc.auxiliary_domains,
delta_fn_left.auxiliary_domains)
self.assertIsInstance(delta_fn_left_disc, pybamm.Multiplication)
self.assertIsInstance(delta_fn_left_disc.left, pybamm.Matrix)
np.testing.assert_array_equal(
delta_fn_left_disc.left.evaluate()[:, 1:], 0)
self.assertEqual(delta_fn_left_disc.shape, y.shape)
# Right
self.assertEqual(delta_fn_right_disc.domain, delta_fn_right.domain)
self.assertEqual(delta_fn_right_disc.auxiliary_domains,
delta_fn_right.auxiliary_domains)
self.assertIsInstance(delta_fn_right_disc, pybamm.Multiplication)
self.assertIsInstance(delta_fn_right_disc.left, pybamm.Matrix)
np.testing.assert_array_equal(
delta_fn_right_disc.left.evaluate()[:, :-1], 0)
self.assertEqual(delta_fn_right_disc.shape, y.shape)
# Value tests
# Delta function should integrate to the same thing as variable
var_disc = disc.process_symbol(var)
x = pybamm.standard_spatial_vars.x_n
delta_fn_int_disc = disc.process_symbol(
pybamm.Integral(delta_fn_left, x))
np.testing.assert_array_equal(
var_disc.evaluate(y=y) * mesh["negative electrode"][0].edges[-1],
np.sum(delta_fn_int_disc.evaluate(y=y)),
)
def test_definite_integral(self):
mesh = get_2p1d_mesh_for_testing()
spatial_methods = {
"macroscale": pybamm.FiniteVolume(),
"current collector": pybamm.ScikitFiniteElement(),
}
disc = pybamm.Discretisation(mesh, spatial_methods)
var = pybamm.Variable("var", domain="current collector")
y = pybamm.SpatialVariable("y", ["current collector"])
z = pybamm.SpatialVariable("z", ["current collector"])
integral_eqn = pybamm.Integral(var, [y, z])
disc.set_variable_slices([var])
integral_eqn_disc = disc.process_symbol(integral_eqn)
y_test = 6 * np.ones(mesh["current collector"][0].npts)
fem_mesh = mesh["current collector"][0]
ly = fem_mesh.coordinates[0, -1]
lz = fem_mesh.coordinates[1, -1]
np.testing.assert_array_almost_equal(
integral_eqn_disc.evaluate(None, y_test), 6 * ly * lz)
def test_symbol_new_copy(self):
a = pybamm.Scalar(0)
b = pybamm.Scalar(1)
v_n = pybamm.Variable("v", "negative electrode")
x_n = pybamm.standard_spatial_vars.x_n
v_s = pybamm.Variable("v", "separator")
vec = pybamm.Vector(np.array([1, 2, 3, 4, 5]))
mesh = get_mesh_for_testing()
for symbol in [
a + b,
a - b,
a * b,
a / b,
a**b,
-a,
abs(a),
pybamm.Function(np.sin, a),
pybamm.FunctionParameter("function", {"a": a}),
pybamm.grad(v_n),
pybamm.div(pybamm.grad(v_n)),
pybamm.Integral(a, pybamm.t),
pybamm.IndefiniteIntegral(v_n, x_n),
pybamm.BackwardIndefiniteIntegral(v_n, x_n),
pybamm.BoundaryValue(v_n, "right"),
pybamm.BoundaryGradient(v_n, "right"),
pybamm.PrimaryBroadcast(a, "domain"),
pybamm.SecondaryBroadcast(v_n, "current collector"),
pybamm.FullBroadcast(a, "domain",
{"secondary": "other domain"}),
pybamm.Concatenation(v_n, v_s),
pybamm.NumpyConcatenation(a, b, v_s),
pybamm.DomainConcatenation([v_n, v_s], mesh),
pybamm.Parameter("param"),
pybamm.InputParameter("param"),
pybamm.StateVector(slice(0, 56)),
pybamm.Matrix(np.ones((50, 40))),
pybamm.SpatialVariable("x", ["negative electrode"]),
pybamm.t,
pybamm.Index(vec, 1),
]:
self.assertEqual(symbol.id, symbol.new_copy().id)
def z_average(symbol):
"""
convenience function for creating an average in the z-direction.
Parameters
----------
symbol : :class:`pybamm.Symbol`
The function to be averaged
Returns
-------
:class:`Symbol`
the new averaged symbol
"""
# Can't take average if the symbol evaluates on edges
if symbol.evaluates_on_edges("primary"):
raise ValueError("Can't take the z-average of a symbol that evaluates on edges")
# Symbol must have domain [] or ["current collector"]
if symbol.domain not in [[], ["current collector"]]:
raise pybamm.DomainError(
"""z-average only implemented in the 'current collector' domain,
but symbol has domains {}""".format(
symbol.domain
)
)
# If symbol doesn't have a domain, its average value is itself
if symbol.domain == []:
new_symbol = symbol.new_copy()
new_symbol.parent = None
return new_symbol
# If symbol is a Broadcast, its average value is its child
elif isinstance(symbol, pybamm.Broadcast):
return symbol.orphans[0]
# Otherwise, use Integral to calculate average value
else:
# We compute the length as Integral(1, z) as this will be easier to identify
# for simplifications later on and it gives the correct behaviour when using
# ZeroDimensionalSpatialMethod
z = pybamm.standard_spatial_vars.z
v = pybamm.ones_like(symbol)
l = pybamm.Integral(v, z)
return Integral(symbol, z) / l
def test_process_symbol(self):
parameter_values = pybamm.ParameterValues({"a": 4, "b": 2, "c": 3})
# process parameter
a = pybamm.Parameter("a")
processed_a = parameter_values.process_symbol(a)
self.assertIsInstance(processed_a, pybamm.Scalar)
self.assertEqual(processed_a.value, 4)
# process binary operation
var = pybamm.Variable("var")
add = a + var
processed_add = parameter_values.process_symbol(add)
self.assertIsInstance(processed_add, pybamm.Addition)
self.assertIsInstance(processed_add.children[0], pybamm.Scalar)
self.assertIsInstance(processed_add.children[1], pybamm.Variable)
self.assertEqual(processed_add.children[0].value, 4)
b = pybamm.Parameter("b")
add = a + b
processed_add = parameter_values.process_symbol(add)
self.assertIsInstance(processed_add, pybamm.Scalar)
self.assertEqual(processed_add.value, 6)
scal = pybamm.Scalar(34)
mul = a * scal
processed_mul = parameter_values.process_symbol(mul)
self.assertIsInstance(processed_mul, pybamm.Scalar)
self.assertEqual(processed_mul.value, 136)
# process integral
aa = pybamm.Parameter("a", domain=["negative electrode"])
x = pybamm.SpatialVariable("x", domain=["negative electrode"])
integ = pybamm.Integral(aa, x)
processed_integ = parameter_values.process_symbol(integ)
self.assertIsInstance(processed_integ, pybamm.Integral)
self.assertIsInstance(processed_integ.children[0], pybamm.Scalar)
self.assertEqual(processed_integ.children[0].value, 4)
self.assertEqual(processed_integ.integration_variable[0].id, x.id)
# process unary operation
v = pybamm.Variable("v", domain="test")
grad = pybamm.Gradient(v)
processed_grad = parameter_values.process_symbol(grad)
self.assertIsInstance(processed_grad, pybamm.Gradient)
self.assertIsInstance(processed_grad.children[0], pybamm.Variable)
# process delta function
aa = pybamm.Parameter("a")
delta_aa = pybamm.DeltaFunction(aa, "left", "some domain")
processed_delta_aa = parameter_values.process_symbol(delta_aa)
self.assertIsInstance(processed_delta_aa, pybamm.DeltaFunction)
self.assertEqual(processed_delta_aa.side, "left")
processed_a = processed_delta_aa.children[0]
self.assertIsInstance(processed_a, pybamm.Scalar)
self.assertEqual(processed_a.value, 4)
# process boundary operator (test for BoundaryValue)
aa = pybamm.Parameter("a", domain=["negative electrode"])
x = pybamm.SpatialVariable("x", domain=["negative electrode"])
boundary_op = pybamm.BoundaryValue(aa * x, "left")
processed_boundary_op = parameter_values.process_symbol(boundary_op)
self.assertIsInstance(processed_boundary_op, pybamm.BoundaryOperator)
processed_a = processed_boundary_op.children[0].children[0]
processed_x = processed_boundary_op.children[0].children[1]
self.assertIsInstance(processed_a, pybamm.Scalar)
self.assertEqual(processed_a.value, 4)
self.assertEqual(processed_x.id, x.id)
# process broadcast
whole_cell = ["negative electrode", "separator", "positive electrode"]
broad = pybamm.PrimaryBroadcast(a, whole_cell)
processed_broad = parameter_values.process_symbol(broad)
self.assertIsInstance(processed_broad, pybamm.Broadcast)
self.assertEqual(processed_broad.domain, whole_cell)
self.assertIsInstance(processed_broad.children[0], pybamm.Scalar)
self.assertEqual(processed_broad.children[0].evaluate(), 4)
# process concatenation
conc = pybamm.concatenation(
pybamm.Vector(np.ones(10), domain="test"),
pybamm.Vector(2 * np.ones(15), domain="test 2"),
)
processed_conc = parameter_values.process_symbol(conc)
self.assertIsInstance(processed_conc.children[0], pybamm.Vector)
self.assertIsInstance(processed_conc.children[1], pybamm.Vector)
np.testing.assert_array_equal(processed_conc.children[0].entries, 1)
np.testing.assert_array_equal(processed_conc.children[1].entries, 2)
# process domain concatenation
c_e_n = pybamm.Variable("c_e_n", ["negative electrode"])
c_e_s = pybamm.Variable("c_e_p", ["separator"])
test_mesh = shared.get_mesh_for_testing()
dom_con = pybamm.DomainConcatenation([a * c_e_n, b * c_e_s], test_mesh)
processed_dom_con = parameter_values.process_symbol(dom_con)
a_proc = processed_dom_con.children[0].children[0]
b_proc = processed_dom_con.children[1].children[0]
self.assertIsInstance(a_proc, pybamm.Scalar)
self.assertIsInstance(b_proc, pybamm.Scalar)
self.assertEqual(a_proc.value, 4)
self.assertEqual(b_proc.value, 2)
# process variable
c = pybamm.Variable("c")
processed_c = parameter_values.process_symbol(c)
self.assertIsInstance(processed_c, pybamm.Variable)
self.assertEqual(processed_c.name, "c")
# process scalar
d = pybamm.Scalar(14)
processed_d = parameter_values.process_symbol(d)
self.assertIsInstance(processed_d, pybamm.Scalar)
self.assertEqual(processed_d.value, 14)
# process array types
e = pybamm.Vector(np.ones(4))
processed_e = parameter_values.process_symbol(e)
self.assertIsInstance(processed_e, pybamm.Vector)
np.testing.assert_array_equal(processed_e.evaluate(), np.ones((4, 1)))
f = pybamm.Matrix(np.ones((5, 6)))
processed_f = parameter_values.process_symbol(f)
self.assertIsInstance(processed_f, pybamm.Matrix)
np.testing.assert_array_equal(processed_f.evaluate(), np.ones((5, 6)))
# process statevector
g = pybamm.StateVector(slice(0, 10))
processed_g = parameter_values.process_symbol(g)
self.assertIsInstance(processed_g, pybamm.StateVector)
np.testing.assert_array_equal(
processed_g.evaluate(y=np.ones(10)), np.ones((10, 1))
)
# not implemented
sym = pybamm.Symbol("sym")
with self.assertRaises(NotImplementedError):
parameter_values.process_symbol(sym)
# not found
with self.assertRaises(KeyError):
x = pybamm.Parameter("x")
parameter_values.process_symbol(x)
def test_integral(self):
# time integral
a = pybamm.Symbol("a")
t = pybamm.t
inta = pybamm.Integral(a, t)
self.assertEqual(inta.name, "integral dtime")
# self.assertTrue(inta.definite)
self.assertEqual(inta.children[0].name, a.name)
self.assertEqual(inta.integration_variable[0], t)
self.assertEqual(inta.domain, [])
# space integral
a = pybamm.Symbol("a", domain=["negative electrode"])
x = pybamm.SpatialVariable("x", ["negative electrode"])
inta = pybamm.Integral(a, x)
self.assertEqual(inta.name, "integral dx ['negative electrode']")
self.assertEqual(inta.children[0].name, a.name)
self.assertEqual(inta.integration_variable[0], x)
self.assertEqual(inta.domain, [])
self.assertEqual(inta.auxiliary_domains, {})
# space integral with secondary domain
a_sec = pybamm.Symbol(
"a",
domain=["negative electrode"],
auxiliary_domains={"secondary": "current collector"},
)
x = pybamm.SpatialVariable("x", ["negative electrode"])
inta_sec = pybamm.Integral(a_sec, x)
self.assertEqual(inta_sec.domain, ["current collector"])
self.assertEqual(inta_sec.auxiliary_domains, {})
# space integral with secondary domain
a_tert = pybamm.Symbol(
"a",
domain=["negative electrode"],
auxiliary_domains={
"secondary": "current collector",
"tertiary": "some extra domain",
},
)
x = pybamm.SpatialVariable("x", ["negative electrode"])
inta_tert = pybamm.Integral(a_tert, x)
self.assertEqual(inta_tert.domain, ["current collector"])
self.assertEqual(inta_tert.auxiliary_domains,
{"secondary": ["some extra domain"]})
# space integral over two variables
b = pybamm.Symbol("b", domain=["current collector"])
y = pybamm.SpatialVariable("y", ["current collector"])
z = pybamm.SpatialVariable("z", ["current collector"])
inta = pybamm.Integral(b, [y, z])
self.assertEqual(inta.name, "integral dy dz ['current collector']")
self.assertEqual(inta.children[0].name, b.name)
self.assertEqual(inta.integration_variable[0], y)
self.assertEqual(inta.integration_variable[1], z)
self.assertEqual(inta.domain, [])
# Indefinite
inta = pybamm.IndefiniteIntegral(a, x)
self.assertEqual(inta.name,
"a integrated w.r.t x on ['negative electrode']")
self.assertEqual(inta.children[0].name, a.name)
self.assertEqual(inta.integration_variable[0], x)
self.assertEqual(inta.domain, ["negative electrode"])
inta_sec = pybamm.IndefiniteIntegral(a_sec, x)
self.assertEqual(inta_sec.domain, ["negative electrode"])
self.assertEqual(inta_sec.auxiliary_domains,
{"secondary": ["current collector"]})
# expected errors
a = pybamm.Symbol("a", domain=["negative electrode"])
x = pybamm.SpatialVariable("x", ["separator"])
y = pybamm.Variable("y")
z = pybamm.SpatialVariable("z", ["negative electrode"])
with self.assertRaises(pybamm.DomainError):
pybamm.Integral(a, x)
with self.assertRaises(ValueError):
pybamm.Integral(a, y)
def test_symbol_simplify(self):
a = pybamm.Scalar(0)
b = pybamm.Scalar(1)
c = pybamm.Parameter("c")
d = pybamm.Scalar(-1)
e = pybamm.Scalar(2)
g = pybamm.Variable("g")
# negate
self.assertIsInstance((-a).simplify(), pybamm.Scalar)
self.assertEqual((-a).simplify().evaluate(), 0)
self.assertIsInstance((-b).simplify(), pybamm.Scalar)
self.assertEqual((-b).simplify().evaluate(), -1)
# absolute value
self.assertIsInstance((abs(a)).simplify(), pybamm.Scalar)
self.assertEqual((abs(a)).simplify().evaluate(), 0)
self.assertIsInstance((abs(d)).simplify(), pybamm.Scalar)
self.assertEqual((abs(d)).simplify().evaluate(), 1)
# function
def sin(x):
return math.sin(x)
f = pybamm.Function(sin, b)
self.assertIsInstance((f).simplify(), pybamm.Scalar)
self.assertEqual((f).simplify().evaluate(), math.sin(1))
def myfunction(x, y):
return x * y
f = pybamm.Function(myfunction, a, b)
self.assertIsInstance((f).simplify(), pybamm.Scalar)
self.assertEqual((f).simplify().evaluate(), 0)
# FunctionParameter
f = pybamm.FunctionParameter("function", b)
self.assertIsInstance((f).simplify(), pybamm.FunctionParameter)
self.assertEqual((f).simplify().children[0].id, b.id)
f = pybamm.FunctionParameter("function", a, b)
self.assertIsInstance((f).simplify(), pybamm.FunctionParameter)
self.assertEqual((f).simplify().children[0].id, a.id)
self.assertEqual((f).simplify().children[1].id, b.id)
# Gradient
self.assertIsInstance((pybamm.grad(a)).simplify(), pybamm.Scalar)
self.assertEqual((pybamm.grad(a)).simplify().evaluate(), 0)
v = pybamm.Variable("v")
self.assertIsInstance((pybamm.grad(v)).simplify(), pybamm.Gradient)
# Divergence
self.assertIsInstance((pybamm.div(a)).simplify(), pybamm.Scalar)
self.assertEqual((pybamm.div(a)).simplify().evaluate(), 0)
self.assertIsInstance((pybamm.div(v)).simplify(), pybamm.Divergence)
# Integral
self.assertIsInstance(
(pybamm.Integral(a, pybamm.t)).simplify(), pybamm.Integral
)
# BoundaryValue
v_neg = pybamm.Variable("v", domain=["negative electrode"])
self.assertIsInstance(
(pybamm.boundary_value(v_neg, "right")).simplify(), pybamm.BoundaryValue
)
# Delta function
self.assertIsInstance(
(pybamm.DeltaFunction(v_neg, "right", "domain")).simplify(),
pybamm.DeltaFunction,
)
# addition
self.assertIsInstance((a + b).simplify(), pybamm.Scalar)
self.assertEqual((a + b).simplify().evaluate(), 1)
self.assertIsInstance((b + b).simplify(), pybamm.Scalar)
self.assertEqual((b + b).simplify().evaluate(), 2)
self.assertIsInstance((b + a).simplify(), pybamm.Scalar)
self.assertEqual((b + a).simplify().evaluate(), 1)
# subtraction
self.assertIsInstance((a - b).simplify(), pybamm.Scalar)
self.assertEqual((a - b).simplify().evaluate(), -1)
self.assertIsInstance((b - b).simplify(), pybamm.Scalar)
self.assertEqual((b - b).simplify().evaluate(), 0)
self.assertIsInstance((b - a).simplify(), pybamm.Scalar)
self.assertEqual((b - a).simplify().evaluate(), 1)
# addition and subtraction with matrix zero
v = pybamm.Vector(np.zeros((10, 1)))
self.assertIsInstance((b + v).simplify(), pybamm.Array)
np.testing.assert_array_equal((b + v).simplify().evaluate(), np.ones((10, 1)))
self.assertIsInstance((v + b).simplify(), pybamm.Array)
np.testing.assert_array_equal((v + b).simplify().evaluate(), np.ones((10, 1)))
self.assertIsInstance((b - v).simplify(), pybamm.Array)
np.testing.assert_array_equal((b - v).simplify().evaluate(), np.ones((10, 1)))
self.assertIsInstance((v - b).simplify(), pybamm.Array)
np.testing.assert_array_equal((v - b).simplify().evaluate(), -np.ones((10, 1)))
# multiplication
self.assertIsInstance((a * b).simplify(), pybamm.Scalar)
self.assertEqual((a * b).simplify().evaluate(), 0)
self.assertIsInstance((b * a).simplify(), pybamm.Scalar)
self.assertEqual((b * a).simplify().evaluate(), 0)
self.assertIsInstance((b * b).simplify(), pybamm.Scalar)
self.assertEqual((b * b).simplify().evaluate(), 1)
self.assertIsInstance((a * a).simplify(), pybamm.Scalar)
self.assertEqual((a * a).simplify().evaluate(), 0)
# test when other node is a parameter
self.assertIsInstance((a + c).simplify(), pybamm.Parameter)
self.assertIsInstance((c + a).simplify(), pybamm.Parameter)
self.assertIsInstance((c + b).simplify(), pybamm.Addition)
self.assertIsInstance((b + c).simplify(), pybamm.Addition)
self.assertIsInstance((a * c).simplify(), pybamm.Scalar)
self.assertEqual((a * c).simplify().evaluate(), 0)
self.assertIsInstance((c * a).simplify(), pybamm.Scalar)
self.assertEqual((c * a).simplify().evaluate(), 0)
self.assertIsInstance((b * c).simplify(), pybamm.Parameter)
self.assertIsInstance((e * c).simplify(), pybamm.Multiplication)
expr = (e * (e * c)).simplify()
self.assertIsInstance(expr, pybamm.Multiplication)
self.assertIsInstance(expr.children[0], pybamm.Scalar)
self.assertIsInstance(expr.children[1], pybamm.Parameter)
expr = (e / (e * c)).simplify()
self.assertIsInstance(expr, pybamm.Division)
self.assertIsInstance(expr.children[0], pybamm.Scalar)
self.assertEqual(expr.children[0].evaluate(), 1.0)
self.assertIsInstance(expr.children[1], pybamm.Parameter)
expr = (e * (e / c)).simplify()
self.assertIsInstance(expr, pybamm.Division)
self.assertIsInstance(expr.children[0], pybamm.Scalar)
self.assertEqual(expr.children[0].evaluate(), 4.0)
self.assertIsInstance(expr.children[1], pybamm.Parameter)
expr = (e * (c / e)).simplify()
self.assertIsInstance(expr, pybamm.Multiplication)
self.assertIsInstance(expr.children[0], pybamm.Scalar)
self.assertEqual(expr.children[0].evaluate(), 1.0)
self.assertIsInstance(expr.children[1], pybamm.Parameter)
expr = ((e * c) * (c / e)).simplify()
self.assertIsInstance(expr, pybamm.Multiplication)
self.assertIsInstance(expr.children[0], pybamm.Scalar)
self.assertEqual(expr.children[0].evaluate(), 1.0)
self.assertIsInstance(expr.children[1], pybamm.Multiplication)
self.assertIsInstance(expr.children[1].children[0], pybamm.Parameter)
self.assertIsInstance(expr.children[1].children[1], pybamm.Parameter)
expr = (e + (e + c)).simplify()
self.assertIsInstance(expr, pybamm.Addition)
self.assertIsInstance(expr.children[0], pybamm.Scalar)
self.assertEqual(expr.children[0].evaluate(), 4.0)
self.assertIsInstance(expr.children[1], pybamm.Parameter)
expr = (e + (e - c)).simplify()
self.assertIsInstance(expr, pybamm.Addition)
self.assertIsInstance(expr.children[0], pybamm.Scalar)
self.assertEqual(expr.children[0].evaluate(), 4.0)
self.assertIsInstance(expr.children[1], pybamm.Negate)
self.assertIsInstance(expr.children[1].children[0], pybamm.Parameter)
expr = (e + (g - c)).simplify()
self.assertIsInstance(expr, pybamm.Addition)
self.assertIsInstance(expr.children[0], pybamm.Scalar)
self.assertEqual(expr.children[0].evaluate(), 2.0)
self.assertIsInstance(expr.children[1], pybamm.Subtraction)
self.assertIsInstance(expr.children[1].children[0], pybamm.Variable)
self.assertIsInstance(expr.children[1].children[1], pybamm.Parameter)
expr = ((2 + c) + (c + 2)).simplify()
self.assertIsInstance(expr, pybamm.Addition)
self.assertIsInstance(expr.children[0], pybamm.Scalar)
self.assertEqual(expr.children[0].evaluate(), 4.0)
self.assertIsInstance(expr.children[1], pybamm.Multiplication)
self.assertIsInstance(expr.children[1].children[0], pybamm.Scalar)
self.assertEqual(expr.children[1].children[0].evaluate(), 2)
self.assertIsInstance(expr.children[1].children[1], pybamm.Parameter)
expr = ((-1 + c) - (c + 1) + (c - 1)).simplify()
self.assertIsInstance(expr, pybamm.Addition)
self.assertIsInstance(expr.children[0], pybamm.Scalar)
self.assertEqual(expr.children[0].evaluate(), -3.0)
# check these don't simplify
self.assertIsInstance((c * e).simplify(), pybamm.Multiplication)
self.assertIsInstance((e / c).simplify(), pybamm.Division)
self.assertIsInstance((c).simplify(), pybamm.Parameter)
c1 = pybamm.Parameter("c1")
self.assertIsInstance((c1 * c).simplify(), pybamm.Multiplication)
# should simplify division to multiply
self.assertIsInstance((c / e).simplify(), pybamm.Multiplication)
self.assertIsInstance((c / b).simplify(), pybamm.Parameter)
self.assertIsInstance((c * b).simplify(), pybamm.Parameter)
# negation with parameter
self.assertIsInstance((-c).simplify(), pybamm.Negate)
self.assertIsInstance((a + b + a).simplify(), pybamm.Scalar)
self.assertEqual((a + b + a).simplify().evaluate(), 1)
self.assertIsInstance((b + a + a).simplify(), pybamm.Scalar)
self.assertEqual((b + a + a).simplify().evaluate(), 1)
self.assertIsInstance((a * b * b).simplify(), pybamm.Scalar)
self.assertEqual((a * b * b).simplify().evaluate(), 0)
self.assertIsInstance((b * a * b).simplify(), pybamm.Scalar)
self.assertEqual((b * a * b).simplify().evaluate(), 0)
# power simplification
self.assertIsInstance((c ** a).simplify(), pybamm.Scalar)
self.assertEqual((c ** a).simplify().evaluate(), 1)
self.assertIsInstance((a ** c).simplify(), pybamm.Scalar)
self.assertEqual((a ** c).simplify().evaluate(), 0)
d = pybamm.Scalar(2)
self.assertIsInstance((c ** d).simplify(), pybamm.Power)
# division
self.assertIsInstance((a / b).simplify(), pybamm.Scalar)
self.assertEqual((a / b).simplify().evaluate(), 0)
self.assertIsInstance((b / a).simplify(), pybamm.Scalar)
self.assertEqual((b / a).simplify().evaluate(), np.inf)
self.assertIsInstance((a / a).simplify(), pybamm.Scalar)
self.assertTrue(np.isnan((a / a).simplify().evaluate()))
self.assertIsInstance((b / b).simplify(), pybamm.Scalar)
self.assertEqual((b / b).simplify().evaluate(), 1)
# not implemented for Symbol
sym = pybamm.Symbol("sym")
with self.assertRaises(NotImplementedError):
sym.simplify()
# A + A = 2A (#323)
a = pybamm.Parameter("A")
expr = (a + a).simplify()
self.assertIsInstance(expr, pybamm.Multiplication)
self.assertIsInstance(expr.children[0], pybamm.Scalar)
self.assertEqual(expr.children[0].evaluate(), 2)
self.assertIsInstance(expr.children[1], pybamm.Parameter)
expr = (a + a + a + a).simplify()
self.assertIsInstance(expr, pybamm.Multiplication)
self.assertIsInstance(expr.children[0], pybamm.Scalar)
self.assertEqual(expr.children[0].evaluate(), 4)
self.assertIsInstance(expr.children[1], pybamm.Parameter)
expr = (a - a + a - a + a + a).simplify()
self.assertIsInstance(expr, pybamm.Multiplication)
self.assertIsInstance(expr.children[0], pybamm.Scalar)
self.assertEqual(expr.children[0].evaluate(), 2)
self.assertIsInstance(expr.children[1], pybamm.Parameter)
# A - A = 0 (#323)
expr = (a - a).simplify()
self.assertIsInstance(expr, pybamm.Scalar)
self.assertEqual(expr.evaluate(), 0)
# B - (A+A) = B - 2*A (#323)
expr = (b - (a + a)).simplify()
self.assertIsInstance(expr, pybamm.Addition)
self.assertIsInstance(expr.right, pybamm.Negate)
self.assertIsInstance(expr.right.child, pybamm.Multiplication)
self.assertEqual(expr.right.child.left.id, pybamm.Scalar(2).id)
self.assertEqual(expr.right.child.right.id, a.id)
# B - (1*A + 2*A) = B - 3*A (#323)
expr = (b - (1 * a + 2 * a)).simplify()
self.assertIsInstance(expr, pybamm.Addition)
self.assertIsInstance(expr.right, pybamm.Negate)
self.assertIsInstance(expr.right.child, pybamm.Multiplication)
self.assertEqual(expr.right.child.left.id, pybamm.Scalar(3).id)
self.assertEqual(expr.right.child.right.id, a.id)
# B - (A + C) = B - (A + C) (not B - (A - C))
expr = (b - (a + c)).simplify()
self.assertIsInstance(expr, pybamm.Addition)
self.assertIsInstance(expr.right, pybamm.Subtraction)
self.assertEqual(expr.right.left.id, (-a).id)
self.assertEqual(expr.right.right.id, c.id)
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 x_average(symbol):
"""convenience function for creating an average in the x-direction
Parameters
----------
symbol : :class:`pybamm.Symbol`
The function to be averaged
Returns
-------
:class:`Symbol`
the new averaged symbol
"""
# Can't take average if the symbol evaluates on edges
if symbol.evaluates_on_edges("primary"):
raise ValueError(
"Can't take the x-average of a symbol that evaluates on edges")
# If symbol doesn't have a domain, its average value is itself
if symbol.domain in [[], ["current collector"]]:
new_symbol = symbol.new_copy()
new_symbol.parent = None
return new_symbol
# If symbol is a Broadcast, its average value is its child
elif isinstance(symbol, pybamm.Broadcast):
return symbol.orphans[0]
# If symbol is a concatenation of Broadcasts, its average value is its child
elif (isinstance(symbol, pybamm.Concatenation) and all(
isinstance(child, pybamm.Broadcast) for child in symbol.children)
and symbol.domain
== ["negative electrode", "separator", "positive electrode"]):
a, b, c = [orp.orphans[0] for orp in symbol.orphans]
if a.id == b.id == c.id:
return a
else:
geo = pybamm.geometric_parameters
l_n = geo.l_n
l_s = geo.l_s
l_p = geo.l_p
return (l_n * a + l_s * b + l_p * c) / (l_n + l_s + l_p)
# Otherwise, use Integral to calculate average value
else:
geo = pybamm.geometric_parameters
if symbol.domain == ["negative electrode"]:
x = pybamm.standard_spatial_vars.x_n
l = geo.l_n
elif symbol.domain == ["separator"]:
x = pybamm.standard_spatial_vars.x_s
l = geo.l_s
elif symbol.domain == ["positive electrode"]:
x = pybamm.standard_spatial_vars.x_p
l = geo.l_p
elif symbol.domain == [
"negative electrode", "separator", "positive electrode"
]:
x = pybamm.standard_spatial_vars.x
l = pybamm.Scalar(1)
elif symbol.domain == ["negative particle"]:
x = pybamm.standard_spatial_vars.x_n
l = geo.l_n
elif symbol.domain == ["positive particle"]:
x = pybamm.standard_spatial_vars.x_p
l = geo.l_p
else:
x = pybamm.SpatialVariable("x", domain=symbol.domain)
v = pybamm.ones_like(symbol)
l = pybamm.Integral(v, x)
return Integral(symbol, x) / l
def test_integral(self):
# space integral
a = pybamm.Symbol("a", domain=["negative electrode"])
x = pybamm.SpatialVariable("x", ["negative electrode"])
inta = pybamm.Integral(a, x)
self.assertEqual(inta.name, "integral dx ['negative electrode']")
self.assertEqual(inta.children[0].name, a.name)
self.assertEqual(inta.integration_variable[0], x)
self.assertEqual(inta.domain, [])
self.assertEqual(inta.auxiliary_domains, {})
# space integral with secondary domain
a_sec = pybamm.Symbol(
"a",
domain=["negative electrode"],
auxiliary_domains={"secondary": "current collector"},
)
x = pybamm.SpatialVariable("x", ["negative electrode"])
inta_sec = pybamm.Integral(a_sec, x)
self.assertEqual(inta_sec.domain, ["current collector"])
self.assertEqual(inta_sec.auxiliary_domains, {})
# space integral with tertiary domain
a_tert = pybamm.Symbol(
"a",
domain=["negative electrode"],
auxiliary_domains={
"secondary": "current collector",
"tertiary": "some extra domain",
},
)
x = pybamm.SpatialVariable("x", ["negative electrode"])
inta_tert = pybamm.Integral(a_tert, x)
self.assertEqual(inta_tert.domain, ["current collector"])
self.assertEqual(inta_tert.auxiliary_domains,
{"secondary": ["some extra domain"]})
# space integral *in* secondary domain
y = pybamm.SpatialVariable("y", ["current collector"])
# without a tertiary domain
inta_sec_y = pybamm.Integral(a_sec, y)
self.assertEqual(inta_sec_y.domain, ["negative electrode"])
self.assertEqual(inta_sec_y.auxiliary_domains, {})
# with a tertiary domain
inta_tert_y = pybamm.Integral(a_tert, y)
self.assertEqual(inta_tert_y.domain, ["negative electrode"])
self.assertEqual(inta_tert_y.auxiliary_domains,
{"secondary": ["some extra domain"]})
# space integral *in* tertiary domain
z = pybamm.SpatialVariable("z", ["some extra domain"])
inta_tert_z = pybamm.Integral(a_tert, z)
self.assertEqual(inta_tert_z.domain, ["negative electrode"])
self.assertEqual(inta_tert_z.auxiliary_domains,
{"secondary": ["current collector"]})
# space integral over two variables
b = pybamm.Symbol("b", domain=["current collector"])
y = pybamm.SpatialVariable("y", ["current collector"])
z = pybamm.SpatialVariable("z", ["current collector"])
inta = pybamm.Integral(b, [y, z])
self.assertEqual(inta.name, "integral dy dz ['current collector']")
self.assertEqual(inta.children[0].name, b.name)
self.assertEqual(inta.integration_variable[0], y)
self.assertEqual(inta.integration_variable[1], z)
self.assertEqual(inta.domain, [])
# Indefinite
inta = pybamm.IndefiniteIntegral(a, x)
self.assertEqual(inta.name,
"a integrated w.r.t x on ['negative electrode']")
self.assertEqual(inta.children[0].name, a.name)
self.assertEqual(inta.integration_variable[0], x)
self.assertEqual(inta.domain, ["negative electrode"])
inta_sec = pybamm.IndefiniteIntegral(a_sec, x)
self.assertEqual(inta_sec.domain, ["negative electrode"])
self.assertEqual(inta_sec.auxiliary_domains,
{"secondary": ["current collector"]})
# backward indefinite integral
inta = pybamm.BackwardIndefiniteIntegral(a, x)
self.assertEqual(
inta.name,
"a integrated backward w.r.t x on ['negative electrode']")
# expected errors
a = pybamm.Symbol("a", domain=["negative electrode"])
x = pybamm.SpatialVariable("x", ["separator"])
y = pybamm.Variable("y")
z = pybamm.SpatialVariable("z", ["negative electrode"])
with self.assertRaises(pybamm.DomainError):
pybamm.Integral(a, x)
with self.assertRaisesRegex(TypeError, "integration_variable must be"):
pybamm.Integral(a, y)
with self.assertRaisesRegex(
NotImplementedError,
"Indefinite integral only implemeted w.r.t. one variable",
):
pybamm.IndefiniteIntegral(a, [x, y])
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 x_average(symbol):
"""
convenience function for creating an average in the x-direction
Parameters
----------
symbol : :class:`pybamm.Symbol`
The function to be averaged
Returns
-------
:class:`Symbol`
the new averaged symbol
"""
# Can't take average if the symbol evaluates on edges
if symbol.evaluates_on_edges("primary"):
raise ValueError(
"Can't take the x-average of a symbol that evaluates on edges")
# If symbol doesn't have a domain, its average value is itself
if symbol.domain in [[], ["current collector"]]:
new_symbol = symbol.new_copy()
new_symbol.parent = None
return new_symbol
# If symbol is a primary or full broadcast, reduce by one dimension
if isinstance(symbol, (pybamm.PrimaryBroadcast, pybamm.FullBroadcast)):
return symbol.reduce_one_dimension()
# If symbol is a concatenation of Broadcasts, its average value is its child
elif (isinstance(symbol, pybamm.Concatenation) and all(
isinstance(child, pybamm.Broadcast) for child in symbol.children)
and symbol.domain
== ["negative electrode", "separator", "positive electrode"]):
a, b, c = [orp.orphans[0] for orp in symbol.orphans]
geo = pybamm.geometric_parameters
l_n = geo.l_n
l_s = geo.l_s
l_p = geo.l_p
out = (l_n * a + l_s * b + l_p * c) / (l_n + l_s + l_p)
# To respect domains we may need to broadcast the child back out
child = symbol.children[0]
# If symbol being returned doesn't have empty domain, return it
if out.domain != []:
return out
# Otherwise we may need to broadcast it
elif child.auxiliary_domains == {}:
return out
else:
domain = child.auxiliary_domains["secondary"]
if "tertiary" not in child.auxiliary_domains:
return pybamm.PrimaryBroadcast(out, domain)
else:
auxiliary_domains = {
"secondary": child.auxiliary_domains["tertiary"]
}
return pybamm.FullBroadcast(out, domain, auxiliary_domains)
# Otherwise, use Integral to calculate average value
else:
geo = pybamm.geometric_parameters
# Even if domain is "negative electrode", "separator", or
# "positive electrode", and we know l, we still compute it as Integral(1, x)
# as this will be easier to identify for simplifications later on
if symbol.domain == ["negative particle"]:
x = pybamm.standard_spatial_vars.x_n
l = geo.l_n
elif symbol.domain == ["positive particle"]:
x = pybamm.standard_spatial_vars.x_p
l = geo.l_p
else:
x = pybamm.SpatialVariable("x", domain=symbol.domain)
v = pybamm.ones_like(symbol)
l = pybamm.Integral(v, x)
return Integral(symbol, x) / l
def test_definite_integral(self):
# create discretisation
mesh = get_mesh_for_testing(xpts=200, rpts=200)
spatial_methods = {
"macroscale": pybamm.FiniteVolume(),
"negative particle": pybamm.FiniteVolume(),
"positive particle": pybamm.FiniteVolume(),
"current collector": pybamm.ZeroDimensionalMethod(),
}
disc = pybamm.Discretisation(mesh, spatial_methods)
# lengths
ln = mesh["negative electrode"][0].edges[-1]
ls = mesh["separator"][0].edges[-1] - ln
lp = 1 - (ln + ls)
# macroscale variable
var = pybamm.Variable("var",
domain=["negative electrode", "separator"])
x = pybamm.SpatialVariable("x", ["negative electrode", "separator"])
integral_eqn = pybamm.Integral(var, x)
disc.set_variable_slices([var])
integral_eqn_disc = disc.process_symbol(integral_eqn)
combined_submesh = mesh.combine_submeshes("negative electrode",
"separator")
constant_y = np.ones_like(combined_submesh[0].nodes[:, np.newaxis])
self.assertEqual(integral_eqn_disc.evaluate(None, constant_y), ln + ls)
linear_y = combined_submesh[0].nodes
np.testing.assert_array_almost_equal(
integral_eqn_disc.evaluate(None, linear_y), (ln + ls)**2 / 2)
cos_y = np.cos(combined_submesh[0].nodes[:, np.newaxis])
np.testing.assert_array_almost_equal(integral_eqn_disc.evaluate(
None, cos_y),
np.sin(ln + ls),
decimal=4)
# domain not starting at zero
var = pybamm.Variable("var",
domain=["separator", "positive electrode"])
x = pybamm.SpatialVariable("x", ["separator", "positive electrode"])
integral_eqn = pybamm.Integral(var, x)
disc.set_variable_slices([var])
integral_eqn_disc = disc.process_symbol(integral_eqn)
combined_submesh = mesh.combine_submeshes("separator",
"positive electrode")
constant_y = np.ones_like(combined_submesh[0].nodes[:, np.newaxis])
self.assertEqual(integral_eqn_disc.evaluate(None, constant_y), ls + lp)
linear_y = combined_submesh[0].nodes
self.assertAlmostEqual(
integral_eqn_disc.evaluate(None, linear_y)[0][0],
(1 - (ln)**2) / 2)
cos_y = np.cos(combined_submesh[0].nodes[:, np.newaxis])
np.testing.assert_array_almost_equal(integral_eqn_disc.evaluate(
None, cos_y),
np.sin(1) - np.sin(ln),
decimal=4)
# microscale variable
var = pybamm.Variable("var", domain=["negative particle"])
r = pybamm.SpatialVariable("r", ["negative particle"])
integral_eqn = pybamm.Integral(var, r)
disc.set_variable_slices([var])
integral_eqn_disc = disc.process_symbol(integral_eqn)
constant_y = np.ones_like(
mesh["negative particle"][0].nodes[:, np.newaxis])
np.testing.assert_array_almost_equal(
integral_eqn_disc.evaluate(None, constant_y), 2 * np.pi**2)
linear_y = mesh["negative particle"][0].nodes
np.testing.assert_array_almost_equal(integral_eqn_disc.evaluate(
None, linear_y),
4 * np.pi**2 / 3,
decimal=3)
one_over_y = 1 / mesh["negative particle"][0].nodes
np.testing.assert_array_almost_equal(
integral_eqn_disc.evaluate(None, one_over_y), 4 * np.pi**2)