#
# Binary operator classes
#
import numbers
import numpy as np
import sympy
from scipy.sparse import csr_matrix, issparse
import pybamm
def preprocess_binary(left, right):
if isinstance(left, numbers.Number):
left = pybamm.Scalar(left)
if isinstance(right, numbers.Number):
right = pybamm.Scalar(right)
# Check both left and right are pybamm Symbols
if not (isinstance(left, pybamm.Symbol) and isinstance(right, pybamm.Symbol)):
raise NotImplementedError(
"""BinaryOperator not implemented for symbols of type {} and {}""".format(
type(left), type(right)
)
)
# Do some broadcasting in special cases, to avoid having to do this manually
if left.domain != [] and right.domain != [] and left.domain != right.domain:
if left.domain == right.secondary_domain:
left = pybamm.PrimaryBroadcast(left, right.domain)
elif right.domain == left.secondary_domain:
right = pybamm.PrimaryBroadcast(right, left.domain)
return left, right
[docs]class BinaryOperator(pybamm.Symbol):
"""
A node in the expression tree representing a binary operator (e.g. `+`, `*`)
Derived classes will specify the particular operator
**Extends**: :class:`Symbol`
Parameters
----------
name : str
name of the node
left : :class:`Symbol` or :class:`Number`
lhs child node (converted to :class:`Scalar` if Number)
right : :class:`Symbol` or :class:`Number`
rhs child node (converted to :class:`Scalar` if Number)
"""
def __init__(self, name, left, right):
left, right = preprocess_binary(left, right)
domains = self.get_children_domains([left, right])
super().__init__(name, children=[left, right], domains=domains)
self.left = self.children[0]
self.right = self.children[1]
def __str__(self):
"""See :meth:`pybamm.Symbol.__str__()`."""
# Possibly add brackets for clarity
if isinstance(self.left, pybamm.BinaryOperator) and not (
(self.left.name == self.name)
or (self.left.name == "*" and self.name == "/")
or (self.left.name == "+" and self.name == "-")
or self.name == "+"
):
left_str = "({!s})".format(self.left)
else:
left_str = "{!s}".format(self.left)
if isinstance(self.right, pybamm.BinaryOperator) and not (
(self.name == "*" and self.right.name in ["*", "/"]) or self.name == "+"
):
right_str = "({!s})".format(self.right)
else:
right_str = "{!s}".format(self.right)
return "{} {} {}".format(left_str, self.name, right_str)
[docs] def create_copy(self):
"""See :meth:`pybamm.Symbol.new_copy()`."""
# process children
new_left = self.left.new_copy()
new_right = self.right.new_copy()
# make new symbol, ensure domain(s) remain the same
out = self._binary_new_copy(new_left, new_right)
out.copy_domains(self)
return out
def _binary_new_copy(self, left, right):
"""
Default behaviour for new_copy.
This copies the behaviour of `_binary_evaluate`, but since `left` and `right`
are symbols creates a new symbol instead of returning a value.
"""
return self._binary_evaluate(left, right)
[docs] def evaluate(self, t=None, y=None, y_dot=None, inputs=None, known_evals=None):
"""See :meth:`pybamm.Symbol.evaluate()`."""
if known_evals is not None:
id = self.id
try:
return known_evals[id], known_evals
except KeyError:
left, known_evals = self.left.evaluate(t, y, y_dot, inputs, known_evals)
right, known_evals = self.right.evaluate(
t, y, y_dot, inputs, known_evals
)
value = self._binary_evaluate(left, right)
known_evals[id] = value
return value, known_evals
else:
left = self.left.evaluate(t, y, y_dot, inputs)
right = self.right.evaluate(t, y, y_dot, inputs)
return self._binary_evaluate(left, right)
def _evaluate_for_shape(self):
"""See :meth:`pybamm.Symbol.evaluate_for_shape()`."""
left = self.children[0].evaluate_for_shape()
right = self.children[1].evaluate_for_shape()
return self._binary_evaluate(left, right)
def _binary_jac(self, left_jac, right_jac):
"""Calculate the jacobian of a binary operator."""
raise NotImplementedError
def _binary_evaluate(self, left, right):
"""Perform binary operation on nodes 'left' and 'right'."""
raise NotImplementedError(
f"{self.__class__} does not implement _binary_evaluate."
)
def _evaluates_on_edges(self, dimension):
"""See :meth:`pybamm.Symbol._evaluates_on_edges()`."""
return self.left.evaluates_on_edges(dimension) or self.right.evaluates_on_edges(
dimension
)
[docs] def is_constant(self):
"""See :meth:`pybamm.Symbol.is_constant()`."""
return self.left.is_constant() and self.right.is_constant()
def _sympy_operator(self, left, right):
"""Apply appropriate SymPy operators."""
return self._binary_evaluate(left, right)
[docs] def to_equation(self):
"""Convert the node and its subtree into a SymPy equation."""
if self.print_name is not None:
return sympy.Symbol(self.print_name)
else:
child1, child2 = self.children
eq1 = child1.to_equation()
eq2 = child2.to_equation()
return self._sympy_operator(eq1, eq2)
[docs]class Power(BinaryOperator):
"""
A node in the expression tree representing a `**` power operator.
**Extends:** :class:`BinaryOperator`
"""
def __init__(self, left, right):
"""See :meth:`pybamm.BinaryOperator.__init__()`."""
super().__init__("**", left, right)
def _diff(self, variable):
"""See :meth:`pybamm.Symbol._diff()`."""
# apply chain rule and power rule
base, exponent = self.orphans
# derivative if variable is in the base
diff = exponent * (base ** (exponent - 1)) * base.diff(variable)
# derivative if variable is in the exponent (rare, check separately to avoid
# unecessarily big tree)
if any(variable.id == x.id for x in exponent.pre_order()):
diff += (base ** exponent) * pybamm.log(base) * exponent.diff(variable)
return diff
def _binary_jac(self, left_jac, right_jac):
"""See :meth:`pybamm.BinaryOperator._binary_jac()`."""
# apply chain rule and power rule
left, right = self.orphans
if right.evaluates_to_constant_number():
return (right * left ** (right - 1)) * left_jac
elif left.evaluates_to_constant_number():
return (left ** right * pybamm.log(left)) * right_jac
else:
return (left ** (right - 1)) * (
right * left_jac + left * pybamm.log(left) * right_jac
)
def _binary_evaluate(self, left, right):
"""See :meth:`pybamm.BinaryOperator._binary_evaluate()`."""
# don't raise RuntimeWarning for NaNs
with np.errstate(invalid="ignore"):
return left ** right
[docs]class Addition(BinaryOperator):
"""
A node in the expression tree representing an addition operator.
**Extends:** :class:`BinaryOperator`
"""
def __init__(self, left, right):
"""See :meth:`pybamm.BinaryOperator.__init__()`."""
super().__init__("+", left, right)
def _diff(self, variable):
"""See :meth:`pybamm.Symbol._diff()`."""
return self.left.diff(variable) + self.right.diff(variable)
def _binary_jac(self, left_jac, right_jac):
"""See :meth:`pybamm.BinaryOperator._binary_jac()`."""
return left_jac + right_jac
def _binary_evaluate(self, left, right):
"""See :meth:`pybamm.BinaryOperator._binary_evaluate()`."""
return left + right
[docs]class Subtraction(BinaryOperator):
"""
A node in the expression tree representing a subtraction operator.
**Extends:** :class:`BinaryOperator`
"""
def __init__(self, left, right):
"""See :meth:`pybamm.BinaryOperator.__init__()`."""
super().__init__("-", left, right)
def _diff(self, variable):
"""See :meth:`pybamm.Symbol._diff()`."""
return self.left.diff(variable) - self.right.diff(variable)
def _binary_jac(self, left_jac, right_jac):
"""See :meth:`pybamm.BinaryOperator._binary_jac()`."""
return left_jac - right_jac
def _binary_evaluate(self, left, right):
"""See :meth:`pybamm.BinaryOperator._binary_evaluate()`."""
return left - right
[docs]class Multiplication(BinaryOperator):
"""
A node in the expression tree representing a multiplication operator
(Hadamard product). Overloads cases where the "*" operator would usually return a
matrix multiplication (e.g. scipy.sparse.coo.coo_matrix)
**Extends:** :class:`BinaryOperator`
"""
def __init__(self, left, right):
"""See :meth:`pybamm.BinaryOperator.__init__()`."""
super().__init__("*", left, right)
def _diff(self, variable):
"""See :meth:`pybamm.Symbol._diff()`."""
# apply product rule
left, right = self.orphans
return left.diff(variable) * right + left * right.diff(variable)
def _binary_jac(self, left_jac, right_jac):
"""See :meth:`pybamm.BinaryOperator._binary_jac()`."""
# apply product rule
left, right = self.orphans
if left.evaluates_to_constant_number():
return left * right_jac
elif right.evaluates_to_constant_number():
return right * left_jac
else:
return right * left_jac + left * right_jac
def _binary_evaluate(self, left, right):
"""See :meth:`pybamm.BinaryOperator._binary_evaluate()`."""
if issparse(left):
return csr_matrix(left.multiply(right))
elif issparse(right):
# Hadamard product is commutative, so we can switch right and left
return csr_matrix(right.multiply(left))
else:
return left * right
[docs]class MatrixMultiplication(BinaryOperator):
"""
A node in the expression tree representing a matrix multiplication operator.
**Extends:** :class:`BinaryOperator`
"""
def __init__(self, left, right):
"""See :meth:`pybamm.BinaryOperator.__init__()`."""
super().__init__("@", left, right)
[docs] def diff(self, variable):
"""See :meth:`pybamm.Symbol.diff()`."""
# We shouldn't need this
raise NotImplementedError(
"diff not implemented for symbol of type 'MatrixMultiplication'"
)
def _binary_jac(self, left_jac, right_jac):
"""See :meth:`pybamm.BinaryOperator._binary_jac()`."""
# We only need the case where left is an array and right
# is a (slice of a) state vector, e.g. for discretised spatial
# operators of the form D @ u (also catch cases of (-D) @ u)
left, right = self.orphans
if isinstance(left, pybamm.Array) or (
isinstance(left, pybamm.Negate) and isinstance(left.child, pybamm.Array)
):
left = pybamm.Matrix(csr_matrix(left.evaluate()))
return left @ right_jac
else:
raise NotImplementedError(
"""jac of 'MatrixMultiplication' is only
implemented for left of type 'pybamm.Array',
not {}""".format(
left.__class__
)
)
def _binary_evaluate(self, left, right):
"""See :meth:`pybamm.BinaryOperator._binary_evaluate()`."""
return left @ right
def _sympy_operator(self, left, right):
"""Override :meth:`pybamm.BinaryOperator._sympy_operator`"""
left = sympy.Matrix(left)
right = sympy.Matrix(right)
return left * right
[docs]class Division(BinaryOperator):
"""
A node in the expression tree representing a division operator.
**Extends:** :class:`BinaryOperator`
"""
def __init__(self, left, right):
"""See :meth:`pybamm.BinaryOperator.__init__()`."""
super().__init__("/", left, right)
def _diff(self, variable):
"""See :meth:`pybamm.Symbol._diff()`."""
# apply quotient rule
top, bottom = self.orphans
return (top.diff(variable) * bottom - top * bottom.diff(variable)) / bottom ** 2
def _binary_jac(self, left_jac, right_jac):
"""See :meth:`pybamm.BinaryOperator._binary_jac()`."""
# apply quotient rule
left, right = self.orphans
if left.evaluates_to_constant_number():
return -left / right ** 2 * right_jac
elif right.evaluates_to_constant_number():
return left_jac / right
else:
return (right * left_jac - left * right_jac) / right ** 2
def _binary_evaluate(self, left, right):
"""See :meth:`pybamm.BinaryOperator._binary_evaluate()`."""
if issparse(left):
return csr_matrix(left.multiply(1 / right))
else:
return left / right
[docs]class Inner(BinaryOperator):
"""
A node in the expression tree which represents the inner (or dot) product. This
operator should be used to take the inner product of two mathematical vectors
(as opposed to the computational vectors arrived at post-discretisation) of the
form v = v_x e_x + v_y e_y + v_z e_z where v_x, v_y, v_z are scalars
and e_x, e_y, e_z are x-y-z-directional unit vectors. For v and w mathematical
vectors, inner product returns v_x * w_x + v_y * w_y + v_z * w_z. In addition,
for some spatial discretisations mathematical vector quantities (such as
i = grad(phi) ) are evaluated on a different part of the grid to mathematical
scalars (e.g. for finite volume mathematical scalars are evaluated on the nodes but
mathematical vectors are evaluated on cell edges). Therefore, inner also transfers
the inner product of the vector onto the scalar part of the grid if required
by a particular discretisation.
**Extends:** :class:`BinaryOperator`
"""
def __init__(self, left, right):
"""See :meth:`pybamm.BinaryOperator.__init__()`."""
super().__init__("inner product", left, right)
def _diff(self, variable):
"""See :meth:`pybamm.Symbol._diff()`."""
# apply product rule
left, right = self.orphans
return left.diff(variable) * right + left * right.diff(variable)
def _binary_jac(self, left_jac, right_jac):
"""See :meth:`pybamm.BinaryOperator._binary_jac()`."""
# apply product rule
left, right = self.orphans
if left.evaluates_to_constant_number():
return left * right_jac
elif right.evaluates_to_constant_number():
return right * left_jac
else:
return right * left_jac + left * right_jac
def _binary_evaluate(self, left, right):
"""See :meth:`pybamm.BinaryOperator._binary_evaluate()`."""
if issparse(left):
return left.multiply(right)
elif issparse(right):
# Hadamard product is commutative, so we can switch right and left
return right.multiply(left)
else:
return left * right
def _binary_new_copy(self, left, right):
"""See :meth:`pybamm.BinaryOperator._binary_new_copy()`."""
return pybamm.inner(left, right)
def _evaluates_on_edges(self, dimension):
"""See :meth:`pybamm.Symbol._evaluates_on_edges()`."""
return False
def inner(left, right):
"""Return inner product of two symbols."""
left, right = preprocess_binary(left, right)
# simplify multiply by scalar zero, being careful about shape
if pybamm.is_scalar_zero(left):
return pybamm.zeros_like(right)
if pybamm.is_scalar_zero(right):
return pybamm.zeros_like(left)
# if one of the children is a zero matrix, we have to be careful about shapes
if pybamm.is_matrix_zero(left) or pybamm.is_matrix_zero(right):
return pybamm.zeros_like(pybamm.Inner(left, right))
# anything multiplied by a scalar one returns itself
if pybamm.is_scalar_one(left):
return right
if pybamm.is_scalar_one(right):
return left
return pybamm.simplify_if_constant(pybamm.Inner(left, right))
class Equality(BinaryOperator):
"""
A node in the expression tree representing an equality comparison between two
nodes. Returns 1 if the two nodes evaluate to the same thing and 0 otherwise.
**Extends:** :class:`BinaryOperator`
"""
def __init__(self, left, right):
"""See :meth:`pybamm.BinaryOperator.__init__()`."""
super().__init__("==", left, right)
def diff(self, variable):
"""See :meth:`pybamm.Symbol.diff()`."""
# Equality should always be multiplied by something else so hopefully don't
# need to worry about shape
return pybamm.Scalar(0)
def _binary_jac(self, left_jac, right_jac):
"""See :meth:`pybamm.BinaryOperator._binary_jac()`."""
# Equality should always be multiplied by something else so hopefully don't
# need to worry about shape
return pybamm.Scalar(0)
def _binary_evaluate(self, left, right):
"""See :meth:`pybamm.BinaryOperator._binary_evaluate()`."""
return int(left == right)
def _binary_new_copy(self, left, right):
"""See :meth:`pybamm.BinaryOperator._binary_new_copy()`."""
return pybamm.Equality(left, right)
[docs]class _Heaviside(BinaryOperator):
"""
A node in the expression tree representing a heaviside step function.
This class is semi-private and should not be called directly, use `EqualHeaviside`
or `NotEqualHeaviside` instead, or `<` or `<=`.
Adding this operation to the rhs or algebraic equations in a model can often cause a
discontinuity in the solution. For the specific cases listed below, this will be
automatically handled by the solver. In the general case, you can explicitly tell
the solver of discontinuities by adding a :class:`Event` object with
:class:`EventType` DISCONTINUITY to the model's list of events.
In the case where the Heaviside function is of the form `pybamm.t < x`, `pybamm.t <=
x`, `x < pybamm.t`, or `x <= pybamm.t`, where `x` is any constant equation, this
DISCONTINUITY event will automatically be added by the solver.
**Extends:** :class:`BinaryOperator`
"""
def __init__(self, name, left, right):
"""See :meth:`pybamm.BinaryOperator.__init__()`."""
super().__init__(name, left, right)
[docs] def diff(self, variable):
"""See :meth:`pybamm.Symbol.diff()`."""
# Heaviside should always be multiplied by something else so hopefully don't
# need to worry about shape
return pybamm.Scalar(0)
def _binary_jac(self, left_jac, right_jac):
"""See :meth:`pybamm.BinaryOperator._binary_jac()`."""
# Heaviside should always be multiplied by something else so hopefully don't
# need to worry about shape
return pybamm.Scalar(0)
[docs]class EqualHeaviside(_Heaviside):
"""A heaviside function with equality (return 1 when left = right)"""
def __init__(self, left, right):
"""See :meth:`pybamm.BinaryOperator.__init__()`."""
super().__init__("<=", left, right)
def __str__(self):
"""See :meth:`pybamm.Symbol.__str__()`."""
return "{!s} <= {!s}".format(self.left, self.right)
def _binary_evaluate(self, left, right):
"""See :meth:`pybamm.BinaryOperator._binary_evaluate()`."""
# don't raise RuntimeWarning for NaNs
with np.errstate(invalid="ignore"):
return left <= right
[docs]class NotEqualHeaviside(_Heaviside):
"""A heaviside function without equality (return 0 when left = right)"""
def __init__(self, left, right):
super().__init__("<", left, right)
def __str__(self):
"""See :meth:`pybamm.Symbol.__str__()`."""
return "{!s} < {!s}".format(self.left, self.right)
def _binary_evaluate(self, left, right):
"""See :meth:`pybamm.BinaryOperator._binary_evaluate()`."""
# don't raise RuntimeWarning for NaNs
with np.errstate(invalid="ignore"):
return left < right
[docs]class Modulo(BinaryOperator):
"""Calculates the remainder of an integer division."""
def __init__(self, left, right):
super().__init__("%", left, right)
def _diff(self, variable):
"""See :meth:`pybamm.Symbol._diff()`."""
# apply chain rule and power rule
left, right = self.orphans
# derivative if variable is in the base
diff = left.diff(variable)
# derivative if variable is in the right term (rare, check separately to avoid
# unecessarily big tree)
if any(variable.id == x.id for x in right.pre_order()):
diff += -pybamm.Floor(left / right) * right.diff(variable)
return diff
def _binary_jac(self, left_jac, right_jac):
"""See :meth:`pybamm.BinaryOperator._binary_jac()`."""
# apply chain rule and power rule
left, right = self.orphans
if right.evaluates_to_constant_number():
return left_jac
elif left.evaluates_to_constant_number():
return -right_jac * pybamm.Floor(left / right)
else:
return left_jac - right_jac * pybamm.Floor(left / right)
def __str__(self):
"""See :meth:`pybamm.Symbol.__str__()`."""
return "{!s} mod {!s}".format(self.left, self.right)
def _binary_evaluate(self, left, right):
"""See :meth:`pybamm.BinaryOperator._binary_evaluate()`."""
return left % right
[docs]class Minimum(BinaryOperator):
"""Returns the smaller of two objects."""
def __init__(self, left, right):
super().__init__("minimum", left, right)
def __str__(self):
"""See :meth:`pybamm.Symbol.__str__()`."""
return "minimum({!s}, {!s})".format(self.left, self.right)
def _diff(self, variable):
"""See :meth:`pybamm.Symbol._diff()`."""
left, right = self.orphans
return (left <= right) * left.diff(variable) + (left > right) * right.diff(
variable
)
def _binary_jac(self, left_jac, right_jac):
"""See :meth:`pybamm.BinaryOperator._binary_jac()`."""
left, right = self.orphans
return (left <= right) * left_jac + (left > right) * right_jac
def _binary_evaluate(self, left, right):
"""See :meth:`pybamm.BinaryOperator._binary_evaluate()`."""
# don't raise RuntimeWarning for NaNs
return np.minimum(left, right)
def _binary_new_copy(self, left, right):
"""See :meth:`pybamm.BinaryOperator._binary_new_copy()`."""
return pybamm.minimum(left, right)
[docs]class Maximum(BinaryOperator):
"""Returns the greater of two objects."""
def __init__(self, left, right):
super().__init__("maximum", left, right)
def __str__(self):
"""See :meth:`pybamm.Symbol.__str__()`."""
return "maximum({!s}, {!s})".format(self.left, self.right)
def _diff(self, variable):
"""See :meth:`pybamm.Symbol._diff()`."""
left, right = self.orphans
return (left >= right) * left.diff(variable) + (left < right) * right.diff(
variable
)
def _binary_jac(self, left_jac, right_jac):
"""See :meth:`pybamm.BinaryOperator._binary_jac()`."""
left, right = self.orphans
return (left >= right) * left_jac + (left < right) * right_jac
def _binary_evaluate(self, left, right):
"""See :meth:`pybamm.BinaryOperator._binary_evaluate()`."""
# don't raise RuntimeWarning for NaNs
return np.maximum(left, right)
def _binary_new_copy(self, left, right):
"""See :meth:`pybamm.BinaryOperator._binary_new_copy()`."""
return pybamm.maximum(left, right)
def simplify_elementwise_binary_broadcasts(left, right):
left, right = preprocess_binary(left, right)
def unpack_broadcast_recursive(symbol):
if isinstance(symbol, pybamm.Broadcast):
if symbol.child.domain == []:
return symbol.orphans[0]
elif (
isinstance(symbol.child, pybamm.Broadcast)
and symbol.child.broadcasts_to_nodes
):
out = unpack_broadcast_recursive(symbol.orphans[0])
if out.domain == []:
return out
return symbol
# No need to broadcast if the other symbol already has the shape that is being
# broadcasted to
# Do this recursively
if left.domains == right.domains:
if isinstance(left, pybamm.Broadcast) and left.broadcasts_to_nodes:
left = unpack_broadcast_recursive(left)
elif isinstance(right, pybamm.Broadcast) and right.broadcasts_to_nodes:
right = unpack_broadcast_recursive(right)
return left, right
def simplified_binary_broadcast_concatenation(left, right, operator):
"""
Check if there are concatenations or broadcasts that we can commute the operator
with
"""
# Broadcast commutes with elementwise operators
if isinstance(left, pybamm.Broadcast) and right.domain == []:
return left._unary_new_copy(operator(left.orphans[0], right))
elif isinstance(right, pybamm.Broadcast) and left.domain == []:
return right._unary_new_copy(operator(left, right.orphans[0]))
# Concatenation commutes with elementwise operators
# If one of the sides is constant then commute concatenation with the operator
# Don't do this for ConcatenationVariable objects as these will
# be simplified differently later on
if isinstance(left, pybamm.Concatenation) and not isinstance(
left, pybamm.ConcatenationVariable
):
if right.evaluates_to_constant_number():
return left._concatenation_new_copy(
[operator(child, right) for child in left.orphans]
)
elif (
isinstance(right, pybamm.Concatenation)
and not any(
isinstance(child, (pybamm.Variable, pybamm.StateVector))
for child in right.children
)
and (
all(child.is_constant() for child in left.children)
or all(child.is_constant() for child in right.children)
)
):
return left._concatenation_new_copy(
[
operator(left_child, right_child)
for left_child, right_child in zip(left.orphans, right.orphans)
]
)
if isinstance(right, pybamm.Concatenation) and not isinstance(
right, pybamm.ConcatenationVariable
):
if left.evaluates_to_constant_number():
return right._concatenation_new_copy(
[operator(left, child) for child in right.orphans]
)
def simplified_power(left, right):
left, right = simplify_elementwise_binary_broadcasts(left, right)
# Check for Concatenations and Broadcasts
out = simplified_binary_broadcast_concatenation(left, right, simplified_power)
if out is not None:
return out
# anything to the power of zero is one
if pybamm.is_scalar_zero(right):
return pybamm.ones_like(left)
# zero to the power of anything is zero
if pybamm.is_scalar_zero(left):
return pybamm.Scalar(0)
# anything to the power of one is itself
if pybamm.is_scalar_one(right):
return left
if isinstance(left, Multiplication):
# Simplify (a * b) ** c to (a ** c) * (b ** c)
# if (a ** c) is constant or (b ** c) is constant
if left.left.is_constant() or left.right.is_constant():
l_left, l_right = left.orphans
new_left = l_left ** right
new_right = l_right ** right
if new_left.is_constant() or new_right.is_constant():
return new_left * new_right
elif isinstance(left, Division):
# Simplify (a / b) ** c to (a ** c) / (b ** c)
# if (a ** c) is constant or (b ** c) is constant
if left.left.is_constant() or left.right.is_constant():
l_left, l_right = left.orphans
new_left = l_left ** right
new_right = l_right ** right
if new_left.is_constant() or new_right.is_constant():
return new_left / new_right
return pybamm.simplify_if_constant(pybamm.Power(left, right))
def simplified_addition(left, right):
"""
Note
----
We check for scalars first, then matrices. This is because
(Zero Matrix) + (Zero Scalar)
should return (Zero Matrix), not (Zero Scalar).
"""
left, right = simplify_elementwise_binary_broadcasts(left, right)
# Check for Concatenations and Broadcasts
out = simplified_binary_broadcast_concatenation(left, right, simplified_addition)
if out is not None:
return out
# anything added by a scalar zero returns the other child
elif pybamm.is_scalar_zero(left):
return right
elif pybamm.is_scalar_zero(right):
return left
# Check matrices after checking scalars
elif pybamm.is_matrix_zero(left):
if right.evaluates_to_number():
return right * pybamm.ones_like(left)
# If left object is zero and has size smaller than or equal to right object in
# all dimensions, we can safely return the right object. For example, adding a
# zero vector a matrix, we can just return the matrix.
# When checking evaluation on edges, check dimensions of left object only
elif all(
left_dim_size <= right_dim_size
for left_dim_size, right_dim_size in zip(
left.shape_for_testing, right.shape_for_testing
)
) and all(
left.evaluates_on_edges(dim) == right.evaluates_on_edges(dim)
for dim in left.domains.keys()
):
return right
elif pybamm.is_matrix_zero(right):
if left.evaluates_to_number():
return left * pybamm.ones_like(right)
# See comment above
elif all(
left_dim_size >= right_dim_size
for left_dim_size, right_dim_size in zip(
left.shape_for_testing, right.shape_for_testing
)
) and all(
left.evaluates_on_edges(dim) == right.evaluates_on_edges(dim)
for dim in left.domains.keys()
):
return left
# Return constant if both sides are constant
if left.is_constant() and right.is_constant():
return pybamm.simplify_if_constant(pybamm.Addition(left, right))
# Simplify A @ c + B @ c to (A + B) @ c if (A + B) is constant
# This is a common construction that appears from discretisation of spatial
# operators
elif (
isinstance(left, MatrixMultiplication)
and isinstance(right, MatrixMultiplication)
and left.right.id == right.right.id
):
l_left, l_right = left.orphans
r_left = right.orphans[0]
new_left = l_left + r_left
if new_left.is_constant():
new_sum = new_left @ l_right
new_sum.copy_domains(pybamm.Addition(left, right))
return new_sum
if isinstance(right, pybamm.Addition) and left.is_constant():
# Simplify a + (b + c) to (a + b) + c if (a + b) is constant
if right.left.is_constant():
r_left, r_right = right.orphans
return (left + r_left) + r_right
# Simplify a + (b + c) to (a + c) + b if (a + c) is constant
elif right.right.is_constant():
r_left, r_right = right.orphans
return (left + r_right) + r_left
elif isinstance(right, pybamm.Subtraction) and left.is_constant():
# Simplify a + (b - c) to (a + b) - c if (a + b) is constant
if right.left.is_constant():
r_left, r_right = right.orphans
return (left + r_left) - r_right
# Simplify a + (b - c) to (a - c) + b if (a - c) is constant
elif right.right.is_constant():
r_left, r_right = right.orphans
return (left - r_right) + r_left
if isinstance(left, pybamm.Addition) and right.is_constant():
# Simplify (a + b) + c to a + (b + c) if (b + c) is constant
if left.right.is_constant():
l_left, l_right = left.orphans
return l_left + (l_right + right)
# Simplify (a + b) + c to (a + c) + b if (a + c) is constant
elif left.left.is_constant():
l_left, l_right = left.orphans
return (l_left + right) + l_right
elif isinstance(left, pybamm.Subtraction) and right.is_constant():
# Simplify (a - b) + c to a + (c - b) if (c - b) is constant
if left.right.is_constant():
l_left, l_right = left.orphans
return l_left + (right - l_right)
# Simplify (a - b) + c to (a + c) - b if (a + c) is constant
elif left.left.is_constant():
l_left, l_right = left.orphans
return (l_left + right) - l_right
return pybamm.simplify_if_constant(pybamm.Addition(left, right))
def simplified_subtraction(left, right):
"""
Note
----
We check for scalars first, then matrices. This is because
(Zero Matrix) - (Zero Scalar)
should return (Zero Matrix), not -(Zero Scalar).
"""
left, right = simplify_elementwise_binary_broadcasts(left, right)
# Check for Concatenations and Broadcasts
out = simplified_binary_broadcast_concatenation(left, right, simplified_subtraction)
if out is not None:
return out
# anything added by a scalar zero returns the other child
if pybamm.is_scalar_zero(left):
return -right
if pybamm.is_scalar_zero(right):
return left
# Check matrices after checking scalars
if pybamm.is_matrix_zero(left):
if right.evaluates_to_number():
return -right * pybamm.ones_like(left)
# See comments in simplified_addition
elif all(
left_dim_size <= right_dim_size
for left_dim_size, right_dim_size in zip(
left.shape_for_testing, right.shape_for_testing
)
) and all(
left.evaluates_on_edges(dim) == right.evaluates_on_edges(dim)
for dim in left.domains.keys()
):
return -right
if pybamm.is_matrix_zero(right):
if left.evaluates_to_number():
return left * pybamm.ones_like(right)
# See comments in simplified_addition
elif all(
left_dim_size >= right_dim_size
for left_dim_size, right_dim_size in zip(
left.shape_for_testing, right.shape_for_testing
)
) and all(
left.evaluates_on_edges(dim) == right.evaluates_on_edges(dim)
for dim in left.domains.keys()
):
return left
# Return constant if both sides are constant
if left.is_constant() and right.is_constant():
return pybamm.simplify_if_constant(Subtraction(left, right))
# a symbol minus itself is 0s of the same shape
if left.id == right.id:
return pybamm.zeros_like(left)
if isinstance(right, pybamm.Addition) and left.is_constant():
# Simplify a - (b + c) to (a - b) - c if (a - b) is constant
if right.left.is_constant():
r_left, r_right = right.orphans
return (left - r_left) - r_right
# Simplify a - (b + c) to (a - c) - b if (a - c) is constant
elif right.right.is_constant():
r_left, r_right = right.orphans
return (left - r_right) - r_left
elif isinstance(right, pybamm.Subtraction) and left.is_constant():
# Simplify a - (b - c) to (a - b) + c if (a - b) is constant
if right.left.is_constant():
r_left, r_right = right.orphans
return (left - r_left) + r_right
# Simplify a - (b - c) to (a + c) - b if (a + c) is constant
elif right.right.is_constant():
r_left, r_right = right.orphans
return (left + r_right) - r_left
if isinstance(left, pybamm.Addition) and right.is_constant():
# Simplify (a + b) - c to a + (b - c) if (b - c) is constant
if left.right.is_constant():
l_left, l_right = left.orphans
return l_left + (l_right - right)
# Simplify (a + b) - c to (a - c) + b if (a - c) is constant
elif left.left.is_constant():
l_left, l_right = left.orphans
return (l_left - right) + l_right
elif isinstance(left, pybamm.Subtraction) and right.is_constant():
# Simplify (a - b) - c to a - (c + b) if (c + b) is constant
if left.right.is_constant():
l_left, l_right = left.orphans
return l_left - (right + l_right)
# Simplify (a - b) - c to (a - c) - b if (a - c) is constant
elif left.left.is_constant():
l_left, l_right = left.orphans
return (l_left - right) - l_right
return pybamm.simplify_if_constant(pybamm.Subtraction(left, right))
def simplified_multiplication(left, right):
left, right = simplify_elementwise_binary_broadcasts(left, right)
# Check for Concatenations and Broadcasts
out = simplified_binary_broadcast_concatenation(
left, right, simplified_multiplication
)
if out is not None:
return out
# simplify multiply by scalar zero, being careful about shape
if pybamm.is_scalar_zero(left):
return pybamm.zeros_like(right)
if pybamm.is_scalar_zero(right):
return pybamm.zeros_like(left)
# if one of the children is a zero matrix, we have to be careful about shapes
if pybamm.is_matrix_zero(left) or pybamm.is_matrix_zero(right):
return pybamm.zeros_like(pybamm.Multiplication(left, right))
# anything multiplied by a scalar one returns itself
if pybamm.is_scalar_one(left):
return right
if pybamm.is_scalar_one(right):
return left
# anything multiplied by a scalar negative one returns negative itself
if pybamm.is_scalar_minus_one(left):
return -right
if pybamm.is_scalar_minus_one(right):
return -left
# Return constant if both sides are constant
if left.is_constant() and right.is_constant():
return pybamm.simplify_if_constant(pybamm.Multiplication(left, right))
# anything multiplied by a matrix one returns itself if
# - the shapes are the same
# - both left and right evaluate on edges, or both evaluate on nodes, in all
# dimensions
# (and possibly more generally, but not implemented here)
try:
if left.shape_for_testing == right.shape_for_testing and all(
left.evaluates_on_edges(dim) == right.evaluates_on_edges(dim)
for dim in left.domains.keys()
):
if pybamm.is_matrix_one(left):
return right
elif pybamm.is_matrix_one(right):
return left
# also check for negative one
if pybamm.is_matrix_minus_one(left):
return -right
elif pybamm.is_matrix_minus_one(right):
return -left
except NotImplementedError:
pass
# Simplify (B @ c) * a to (a * B) @ c if (a * B) is constant
# This is a common construction that appears from discretisation of spatial
# operators
if (
isinstance(left, MatrixMultiplication)
and left.left.is_constant()
and right.is_constant()
and not (right.ndim_for_testing == 2 and right.shape_for_testing[1] > 1)
):
l_left, l_right = left.orphans
new_left = right * l_left
# be careful about domains to avoid weird errors
new_left.clear_domains()
new_mul = new_left @ l_right
# Keep the domain of the old left
new_mul.copy_domains(left)
return new_mul
elif isinstance(left, Multiplication) and right.is_constant():
# Simplify (a * b) * c to (a * c) * b if (a * c) is constant
if left.left.is_constant():
l_left, l_right = left.orphans
return (l_left * right) * l_right
# Simplify (a * b) * c to a * (b * c) if (b * c) is constant
elif left.right.is_constant():
l_left, l_right = left.orphans
return l_left * (l_right * right)
elif isinstance(left, Division) and right.is_constant():
# Simplify (a / b) * c to a * (c / b) if (c / b) is constant
if left.right.is_constant():
l_left, l_right = left.orphans
return l_left * (right / l_right)
# Simplify a * (B @ c) to (a * B) @ c if (a * B) is constant
if (
isinstance(right, MatrixMultiplication)
and right.left.is_constant()
and left.is_constant()
and not (left.ndim_for_testing == 2 and left.shape_for_testing[1] > 1)
):
r_left, r_right = right.orphans
new_left = left * r_left
# be careful about domains to avoid weird errors
new_left.clear_domains()
new_mul = new_left @ r_right
# Keep the domain of the old right
new_mul.copy_domains(right)
return new_mul
elif isinstance(right, Multiplication) and left.is_constant():
# Simplify a * (b * c) to (a * b) * c if (a * b) is constant
if right.left.is_constant():
r_left, r_right = right.orphans
return (left * r_left) * r_right
# Simplify a * (b * c) to (a * c) * b if (a * c) is constant
elif right.right.is_constant():
r_left, r_right = right.orphans
return (left * r_right) * r_left
elif isinstance(right, Division) and left.is_constant():
# Simplify a * (b / c) to (a / c) * b if (a / c) is constant
if right.right.is_constant():
r_left, r_right = right.orphans
return (left / r_right) * r_left
# Simplify a * (b + c) to (a * b) + (a * c) if (a * b) or (a * c) is constant
# This is a common construction that appears from discretisation of spatial
# operators
# Also do this for cases like a * (b @ c + d) where (a * b) is constant
elif isinstance(right, (Addition, Subtraction)):
mul_classes = (
pybamm.Multiplication,
pybamm.MatrixMultiplication,
pybamm.Division,
)
if (
right.left.is_constant()
or right.right.is_constant()
or (isinstance(right.left, mul_classes) and right.left.left.is_constant())
or (isinstance(right.right, mul_classes) and right.right.left.is_constant())
):
r_left, r_right = right.orphans
if (r_left.domain == right.domain or r_left.domain == []) and (
r_right.domain == right.domain or r_right.domain == []
):
if isinstance(right, Addition):
return (left * r_left) + (left * r_right)
elif isinstance(right, Subtraction):
return (left * r_left) - (left * r_right)
# Negation simplifications
if isinstance(left, pybamm.Negate) and isinstance(right, pybamm.Negate):
# Double negation cancels out
return left.orphans[0] * right.orphans[0]
elif isinstance(left, pybamm.Negate) and right.is_constant():
# Simplify (-a) * b to a * (-b) if (-b) is constant
return left.orphans[0] * (-right)
elif isinstance(right, pybamm.Negate) and left.is_constant():
# Simplify a * (-b) to (-a) * b if (-a) is constant
return (-left) * right.orphans[0]
return pybamm.Multiplication(left, right)
def simplified_division(left, right):
left, right = simplify_elementwise_binary_broadcasts(left, right)
# Check for Concatenations and Broadcasts
out = simplified_binary_broadcast_concatenation(left, right, simplified_division)
if out is not None:
return out
# zero divided by anything returns zero (being careful about shape)
if pybamm.is_scalar_zero(left):
return pybamm.zeros_like(right)
# matrix zero divided by anything returns matrix zero (i.e. itself)
if pybamm.is_matrix_zero(left):
return pybamm.zeros_like(pybamm.Division(left, right))
# anything divided by zero raises error
if pybamm.is_scalar_zero(right):
raise ZeroDivisionError
# anything divided by one is itself
if pybamm.is_scalar_one(right):
return left
# a symbol divided by itself is 1s of the same shape
if left.id == right.id:
return pybamm.ones_like(left)
# anything multiplied by a matrix one returns itself if
# - the shapes are the same
# - both left and right evaluate on edges, or both evaluate on nodes, in all
# dimensions
# (and possibly more generally, but not implemented here)
try:
if left.shape_for_testing == right.shape_for_testing and all(
left.evaluates_on_edges(dim) == right.evaluates_on_edges(dim)
for dim in left.domains.keys()
):
if pybamm.is_matrix_one(right):
return left
# also check for negative one
if pybamm.is_matrix_minus_one(right):
return -left
except NotImplementedError:
pass
# Return constant if both sides are constant
if left.is_constant() and right.is_constant():
return pybamm.simplify_if_constant(pybamm.Division(left, right))
# Simplify (B @ c) / a to (B / a) @ c if (B / a) is constant
# This is a common construction that appears from discretisation of averages
elif isinstance(left, MatrixMultiplication) and right.is_constant():
l_left, l_right = left.orphans
new_left = l_left / right
if new_left.is_constant():
# be careful about domains to avoid weird errors
new_left.clear_domains()
new_division = new_left @ l_right
# Keep the domain of the old left
new_division.copy_domains(left)
return new_division
if isinstance(left, Multiplication) and right.is_constant():
# Simplify (a * b) / c to (a / c) * b if (a / c) is constant
if left.left.is_constant():
l_left, l_right = left.orphans
return (l_left / right) * l_right
# Simplify (a * b) / c to a * (b / c) if (b / c) is constant
elif left.right.is_constant():
l_left, l_right = left.orphans
return l_left * (l_right / right)
elif isinstance(left, Division) and right.is_constant():
# Simplify (a / b) / c to (a / c) / b if (a / c) is constant
if left.left.is_constant():
l_left, l_right = left.orphans
return (l_left / right) / l_right
# Simplify (a / b) / c to a / (b * c) if (b * c) is constant
elif left.right.is_constant():
l_left, l_right = left.orphans
return l_left / (l_right * right)
if isinstance(right, Multiplication) and left.is_constant():
# Simplify a / (b * c) to (a / b) / c if (a / b) is constant
if right.left.is_constant():
r_left, r_right = right.orphans
return (left / r_left) / r_right
# Simplify a / (b * c) to (a / c) / b if (a / c) is constant
elif right.right.is_constant():
r_left, r_right = right.orphans
return (left / r_right) / r_left
elif isinstance(right, Division) and left.is_constant():
# Simplify a / (b / c) to (a / b) * c if (a / b) is constant
if right.left.is_constant():
r_left, r_right = right.orphans
return (left / r_left) * r_right
# Simplify a / (b / c) to (a * c) / b if (a * c) is constant
elif right.right.is_constant():
r_left, r_right = right.orphans
return (left * r_right) / r_left
# Negation simplifications
if isinstance(left, pybamm.Negate) and isinstance(right, pybamm.Negate):
# Double negation cancels out
return left.orphans[0] / right.orphans[0]
elif isinstance(left, pybamm.Negate) and right.is_constant():
# Simplify (-a) / b to a / (-b) if (-b) is constant
return left.orphans[0] / (-right)
elif isinstance(right, pybamm.Negate) and left.is_constant():
# Simplify a / (-b) to (-a) / b if (-a) is constant
return (-left) / right.orphans[0]
return pybamm.simplify_if_constant(pybamm.Division(left, right))
def simplified_matrix_multiplication(left, right):
left, right = preprocess_binary(left, right)
if pybamm.is_matrix_zero(left) or pybamm.is_matrix_zero(right):
return pybamm.zeros_like(pybamm.MatrixMultiplication(left, right))
if isinstance(right, Multiplication) and left.is_constant():
# Simplify A @ (b * c) to (A * b) @ c if (A * b) is constant
if right.left.evaluates_to_constant_number():
r_left, r_right = right.orphans
return (left * r_left) @ r_right
# Simplify A @ (b * c) to (A * c) @ b if (A * c) is constant
elif right.right.evaluates_to_constant_number():
r_left, r_right = right.orphans
return (left * r_right) @ r_left
elif isinstance(right, Division) and left.is_constant():
# Simplify A @ (b / c) to (A / c) @ b if (A / c) is constant
if right.right.evaluates_to_constant_number():
r_left, r_right = right.orphans
return (left / r_right) @ r_left
# Simplify A @ (B @ c) to (A @ B) @ c if (A @ B) is constant
# This is a common construction that appears from discretisation of spatial
# operators
if (
isinstance(right, MatrixMultiplication)
and right.left.is_constant()
and left.is_constant()
):
r_left, r_right = right.orphans
new_left = left @ r_left
# be careful about domains to avoid weird errors
new_left.clear_domains()
new_mul = new_left @ r_right
# Keep the domain of the old right
new_mul.copy_domains(right)
return new_mul
# Simplify A @ (b + c) to (A @ b) + (A @ c) if (A @ b) or (A @ c) is constant
# This is a common construction that appears from discretisation of spatial
# operators
# Don't do this if either b or c is a number as this will lead to matmul errors
elif isinstance(right, Addition):
if (right.left.is_constant() or right.right.is_constant()) and not (
right.left.size_for_testing == 1 or right.right.size_for_testing == 1
):
r_left, r_right = right.orphans
return (left @ r_left) + (left @ r_right)
return pybamm.simplify_if_constant(pybamm.MatrixMultiplication(left, right))
[docs]def minimum(left, right):
"""
Returns the smaller of two objects, possibly with a smoothing approximation.
Not to be confused with :meth:`pybamm.min`, which returns min function of child.
"""
k = pybamm.settings.min_smoothing
# Return exact approximation if that is the setting or the outcome is a constant
# (i.e. no need for smoothing)
if k == "exact" or (pybamm.is_constant(left) and pybamm.is_constant(right)):
out = Minimum(left, right)
else:
out = pybamm.softminus(left, right, k)
return pybamm.simplify_if_constant(out)
[docs]def maximum(left, right):
"""
Returns the larger of two objects, possibly with a smoothing approximation.
Not to be confused with :meth:`pybamm.max`, which returns max function of child.
"""
k = pybamm.settings.max_smoothing
# Return exact approximation if that is the setting or the outcome is a constant
# (i.e. no need for smoothing)
if k == "exact" or (pybamm.is_constant(left) and pybamm.is_constant(right)):
out = Maximum(left, right)
else:
out = pybamm.softplus(left, right, k)
return pybamm.simplify_if_constant(out)
[docs]def softminus(left, right, k):
"""
Softplus approximation to the minimum function. k is the smoothing parameter,
set by `pybamm.settings.min_smoothing`. The recommended value is k=10.
"""
return pybamm.log(pybamm.exp(-k * left) + pybamm.exp(-k * right)) / -k
[docs]def softplus(left, right, k):
"""
Softplus approximation to the maximum function. k is the smoothing parameter,
set by `pybamm.settings.max_smoothing`. The recommended value is k=10.
"""
return pybamm.log(pybamm.exp(k * left) + pybamm.exp(k * right)) / k
[docs]def sigmoid(left, right, k):
"""
Sigmoidal approximation to the heaviside function. k is the smoothing parameter,
set by `pybamm.settings.heaviside_smoothing`. The recommended value is k=10.
Note that the concept of deciding which side to pick when left=right does not apply
for this smooth approximation. When left=right, the value is (left+right)/2.
"""
return (1 + pybamm.tanh(k * (right - left))) / 2
[docs]def source(left, right, boundary=False):
"""
A convenience function for creating (part of) an expression tree representing
a source term. This is necessary for spatial methods where the mass matrix
is not the identity (e.g. finite element formulation with piecwise linear
basis functions). The left child is the symbol representing the source term
and the right child is the symbol of the equation variable (currently, the
finite element formulation in PyBaMM assumes all functions are constructed
using the same basis, and the matrix here is constructed accoutning for the
boundary conditions of the right child). The method returns the matrix-vector
product of the mass matrix (adjusted to account for any Dirichlet boundary
conditions imposed the the right symbol) and the discretised left symbol.
Parameters
----------
left : :class:`Symbol`
The left child node, which represents the expression for the source term.
right : :class:`Symbol`
The right child node. This is the symbol whose boundary conditions are
accounted for in the construction of the mass matrix.
boundary : bool, optional
If True, then the mass matrix should is assembled over the boundary,
corresponding to a source term which only acts on the boundary of the
domain. If False (default), the matrix is assembled over the entire domain,
corresponding to a source term in the bulk.
"""
# Broadcast if left is number
if isinstance(left, numbers.Number):
left = pybamm.PrimaryBroadcast(left, "current collector")
if left.domain != ["current collector"] or right.domain != ["current collector"]:
raise pybamm.DomainError(
"""'source' only implemented in the 'current collector' domain,
but symbols have domains {} and {}""".format(
left.domain, right.domain
)
)
if boundary:
return pybamm.BoundaryMass(right) @ left
else:
return pybamm.Mass(right) @ left