Source code for ufl.form

```# -*- coding: utf-8 -*-
"The Form class."

# Copyright (C) 2008-2016 Martin Sandve Alnæs
#
# This file is part of UFL (https://www.fenicsproject.org)
#
#
# Modified by Anders Logg, 2009-2011.
# Modified by Massimiliano Leoni, 2016.
# Modified by Cecile Daversin-Catty, 2018.

from itertools import chain
from collections import defaultdict

from ufl.log import error, warning
from ufl.domain import sort_domains
from ufl.integral import Integral
from ufl.checks import is_scalar_constant_expression
from ufl.equation import Equation
from ufl.core.expr import Expr
from ufl.core.expr import ufl_err_str
from ufl.constantvalue import Zero

# Export list for ufl.classes
__all_classes__ = ["Form"]

# --- The Form class, representing a complete variational form or functional ---

def _sorted_integrals(integrals):
"""Sort integrals by domain id, integral type, subdomain id
for a more stable signature computation."""

# Group integrals in multilevel dict by keys
# [domain][integral_type][subdomain_id]
integrals_dict = defaultdict(
lambda: defaultdict(lambda: defaultdict(list)))
for integral in integrals:
d = integral.ufl_domain()
if d is None:
error(
"Each integral in a form must have a uniquely defined integration domain."
)
it = integral.integral_type()
si = integral.subdomain_id()
integrals_dict[d][it][si] += [integral]

all_integrals = []

# Order integrals canonically to increase signature stability
for d in sort_domains(integrals_dict):
for it in sorted(integrals_dict[d]):  # str is sortable
for si in sorted(
integrals_dict[d][it], key=lambda x: (type(x).__name__, x)
):  # int/str are sortable
unsorted_integrals = integrals_dict[d][it][si]
# TODO: At this point we could order integrals by
#       metadata and integrand, or even add the
#       integrands with the same metadata. This is
#       done in
#       algorithms/domain_analysis.py and would
#       further increase the signature stability.
all_integrals.extend(unsorted_integrals)
# integrals_dict[d][it][si] = unsorted_integrals

return tuple(all_integrals)  # integrals_dict

[docs]class Form(object):
"""Description of a weak form consisting of a sum of integrals over subdomains."""
__slots__ = (
# --- List of Integral objects (a Form is a sum of these Integrals, everything else is derived)
"_integrals",
# --- Internal variables for caching various data
"_integration_domains",
"_domain_numbering",
"_subdomain_data",
"_arguments",
"_coefficients",
"_coefficient_numbering",
"_constant_numbering",
"_constants",
"_hash",
"_signature",
# --- Dict that external frameworks can place framework-specific
#     data in to be carried with the form
#     Never use this internally in ufl!
"_cache",
)

def __init__(self, integrals):
# Basic input checking (further compatibilty analysis happens
# later)
if not all(isinstance(itg, Integral) for itg in integrals):
error("Expecting list of integrals.")

# Store integrals sorted canonically to increase signature
# stability
self._integrals = _sorted_integrals(integrals)

# Internal variables for caching domain data
self._integration_domains = None
self._domain_numbering = None

# Internal variables for caching subdomain data
self._subdomain_data = None

# Internal variables for caching form argument data
self._arguments = None
self._coefficients = None
self._coefficient_numbering = None
self._constant_numbering = None

from ufl.algorithms.analysis import extract_constants
self._constants = extract_constants(self)
self._constant_numbering = dict(
(c, i) for i, c in enumerate(self._constants))

# Internal variables for caching of hash and signature after
# first request
self._hash = None
self._signature = None

# Never use this internally in ufl!
self._cache = {}

# --- Accessor interface ---

[docs]    def integrals(self):
"Return a sequence of all integrals in form."
return self._integrals

[docs]    def integrals_by_type(self, integral_type):
"Return a sequence of all integrals with a particular domain type."
return tuple(integral for integral in self.integrals()
if integral.integral_type() == integral_type)

[docs]    def integrals_by_domain(self, domain):
"Return a sequence of all integrals with a particular integration domain."
return tuple(integral for integral in self.integrals()
if integral.ufl_domain() == domain)

[docs]    def empty(self):
"Returns whether the form has no integrals."
return self.integrals() == ()

[docs]    def ufl_domains(self):
"""Return the geometric integration domains occuring in the form.

NB! This does not include domains of coefficients defined on other meshes.

The return type is a tuple even if only a single domain exists.
"""
if self._integration_domains is None:
self._analyze_domains()
return self._integration_domains

[docs]    def ufl_cell(self):
"""Return the single cell this form is defined on, fails if multiple
cells are found.

"""
return self.ufl_domain().ufl_cell()

[docs]    def ufl_domain(self):
"""Return the single geometric integration domain occuring in the
form.

Fails if multiple domains are found.

NB! This does not include domains of coefficients defined on
other meshes, look at form data for that additional
information.

"""
# Collect all domains
domains = self.ufl_domains()
# Check that all are equal TODO: don't return more than one if
# all are equal?
if not all(domain == domains[0] for domain in domains):
error(
"Calling Form.ufl_domain() is only valid if all integrals share domain."
)

# Return the one and only domain
return domains[0]

[docs]    def geometric_dimension(self):
"Return the geometric dimension shared by all domains and functions in this form."
gdims = tuple(
set(domain.geometric_dimension() for domain in self.ufl_domains()))
if len(gdims) != 1:
error("Expecting all domains and functions in a form "
"to share geometric dimension, got %s." % str(
tuple(sorted(gdims))))
return gdims[0]

[docs]    def domain_numbering(self):
"""Return a contiguous numbering of domains in a mapping
``{domain:number}``."""
if self._domain_numbering is None:
self._analyze_domains()
return self._domain_numbering

[docs]    def subdomain_data(self):
"""Returns a mapping on the form ``{domain:{integral_type:
subdomain_data}}``."""
if self._subdomain_data is None:
self._analyze_subdomain_data()
return self._subdomain_data

[docs]    def max_subdomain_ids(self):
"""Returns a mapping on the form
``{domain:{integral_type:max_subdomain_id}}``."""
if self._max_subdomain_ids is None:
self._analyze_subdomain_data()
return self._max_subdomain_ids

[docs]    def arguments(self):
"Return all ``Argument`` objects found in form."
if self._arguments is None:
self._analyze_form_arguments()
return self._arguments

[docs]    def coefficients(self):
"Return all ``Coefficient`` objects found in form."
if self._coefficients is None:
self._analyze_form_arguments()
return self._coefficients

[docs]    def coefficient_numbering(self):
"""Return a contiguous numbering of coefficients in a mapping
``{coefficient:number}``."""
if self._coefficient_numbering is None:
self._analyze_form_arguments()
return self._coefficient_numbering

[docs]    def constants(self):
return self._constants

[docs]    def constant_numbering(self):
"""Return a contiguous numbering of constants in a mapping
``{constant:number}``."""
return self._constant_numbering

[docs]    def signature(self):
"Signature for use with jit cache (independent of incidental numbering of indices etc.)"
if self._signature is None:
self._compute_signature()
return self._signature

# --- Operator implementations ---

def __hash__(self):
"Hash code for use in dicts (includes incidental numbering of indices etc.)"
if self._hash is None:
self._hash = hash(tuple(hash(itg) for itg in self.integrals()))
return self._hash

def __eq__(self, other):
"""Delayed evaluation of the == operator!

Just 'lhs_form == rhs_form' gives an Equation,
while 'bool(lhs_form == rhs_form)' delegates
to lhs_form.equals(rhs_form).
"""
return Equation(self, other)

def __ne__(self, other):
"Immediate evaluation of the != operator (as opposed to the == operator)."
return not self.equals(other)

[docs]    def equals(self, other):
"Evaluate ``bool(lhs_form == rhs_form)``."
if type(other) != Form:
return False
if len(self._integrals) != len(other._integrals):
return False
if hash(self) != hash(other):
return False
return all(a == b for a, b in zip(self._integrals, other._integrals))

# Ordering of form additions make no difference

if isinstance(other, Form):
# Add integrals from both forms
return Form(list(chain(self.integrals(), other.integrals())))

elif isinstance(other, (int, float)) and other == 0:
# Allow adding 0 or 0.0 as a no-op, needed for sum([a,b])
return self

elif isinstance(
other,
Zero) and not (other.ufl_shape or other.ufl_free_indices):
# Allow adding ufl Zero as a no-op, needed for sum([a,b])
return self

else:
# Let python protocols do their job if we don't handle it
return NotImplemented

def __sub__(self, other):
"Subtract other form from this one."
return self + (-other)

def __rsub__(self, other):
"Subtract this form from other."
return other + (-self)

def __neg__(self):
"""Negate all integrals in form.

This enables the handy "-form" syntax for e.g. the
linearized system (J, -F) from a nonlinear form F."""
return Form([-itg for itg in self.integrals()])

def __rmul__(self, scalar):
"Multiply all integrals in form with constant scalar value."
# This enables the handy "0*form" or "dt*form" syntax
if is_scalar_constant_expression(scalar):
return Form([scalar * itg for itg in self.integrals()])
return NotImplemented

def __mul__(self, coefficient):
"UFL form operator: Take the action of this form on the given coefficient."
if isinstance(coefficient, Expr):
from ufl.formoperators import action
return action(self, coefficient)
return NotImplemented

def __call__(self, *args, **kwargs):
"""UFL form operator: Evaluate form by replacing arguments and
coefficients.

Replaces form.arguments() with given positional arguments in
same number and ordering. Number of positional arguments must
be 0 or equal to the number of Arguments in the form.

The optional keyword argument coefficients can be set to a dict
to replace Coefficients with expressions of matching shapes.

Example:
-------
V = FiniteElement("CG", triangle, 1)
v = TestFunction(V)
u = TrialFunction(V)
f = Coefficient(V)
g = Coefficient(V)
M = a(f, f, coefficients={ g: 1 })

Is equivalent to M == grad(f)**2*dx.

"""
repdict = {}

if args:
arguments = self.arguments()
if len(arguments) != len(args):
error("Need %d arguments to form(), got %d." % (len(arguments),
len(args)))
repdict.update(zip(arguments, args))

coefficients = kwargs.pop("coefficients")
if kwargs:
error("Unknown kwargs %s." % str(list(kwargs)))

if coefficients is not None:
coeffs = self.coefficients()
for f in coefficients:
if f in coeffs:
repdict[f] = coefficients[f]
else:
warning("Coefficient %s is not in form." % ufl_err_str(f))
if repdict:
from ufl.formoperators import replace
return replace(self, repdict)
else:
return self

# "a @ f" notation in python 3.5
__matmul__ = __mul__

# --- String conversion functions, for UI purposes only ---

def __str__(self):
"Compute shorter string representation of form. This can be huge for complicated forms."
# Warning used for making sure we don't use this in the general pipeline:
# warning("Calling str on form is potentially expensive and should be avoided except during debugging.")
# Not caching this because it can be huge
s = "\n  +  ".join(str(itg) for itg in self.integrals())
return s or "<empty Form>"

def __repr__(self):
"Compute repr string of form. This can be huge for complicated forms."
# Warning used for making sure we don't use this in the general pipeline:
# warning("Calling repr on form is potentially expensive and should be avoided except during debugging.")
# Not caching this because it can be huge
itgs = ", ".join(repr(itg) for itg in self.integrals())
r = "Form([" + itgs + "])"
return r

# --- Analysis functions, precomputation and caching of various quantities

def _analyze_domains(self):
from ufl.domain import join_domains, sort_domains

# Collect unique integration domains
integration_domains = join_domains(
[itg.ufl_domain() for itg in self._integrals])

# Make canonically ordered list of the domains
self._integration_domains = sort_domains(integration_domains)

# TODO: Not including domains from coefficients and arguments
# here, may need that later
self._domain_numbering = dict(
(d, i) for i, d in enumerate(self._integration_domains))

def _analyze_subdomain_data(self):
integration_domains = self.ufl_domains()
integrals = self.integrals()

# Make clear data structures to collect subdomain data in
subdomain_data = {}
for domain in integration_domains:
subdomain_data[domain] = {}

for integral in integrals:
# Get integral properties
domain = integral.ufl_domain()
it = integral.integral_type()
sd = integral.subdomain_data()

# Collect subdomain data
data = subdomain_data[domain].get(it)
if data is None:
subdomain_data[domain][it] = sd
elif sd is not None:
if data.ufl_id() != sd.ufl_id():
error(
"Integrals in form have different subdomain_data objects."
)
self._subdomain_data = subdomain_data

def _analyze_form_arguments(self):
"Analyze which Argument and Coefficient objects can be found in the form."
from ufl.algorithms.analysis import extract_arguments_and_coefficients
arguments, coefficients = extract_arguments_and_coefficients(self)

# Define canonical numbering of arguments and coefficients
self._arguments = tuple(
sorted(set(arguments), key=lambda x: x.number()))
self._coefficients = tuple(
sorted(set(coefficients), key=lambda x: x.count()))
self._coefficient_numbering = dict(
(c, i) for i, c in enumerate(self._coefficients))

def _compute_renumbering(self):
# Include integration domains and coefficients in renumbering
dn = self.domain_numbering()
cn = self.coefficient_numbering()
cnstn = self.constant_numbering()
renumbering = {}
renumbering.update(dn)
renumbering.update(cn)
renumbering.update(cnstn)

# Add domains of coefficients, these may include domains not
# among integration domains
k = len(dn)
for c in cn:
d = c.ufl_domain()
if d is not None and d not in renumbering:
renumbering[d] = k
k += 1

# Add domains of arguments, these may include domains not
# among integration domains
for a in self._arguments:
d = a.ufl_function_space().ufl_domain()
if d is not None and d not in renumbering:
renumbering[d] = k
k += 1

# Add domains of constants, these may include domains not
# among integration domains
for c in self._constants:
d = c.ufl_domain()
if d is not None and d not in renumbering:
renumbering[d] = k
k += 1

return renumbering

def _compute_signature(self):
from ufl.algorithms.signature import compute_form_signature
self._signature = compute_form_signature(self,
self._compute_renumbering())

[docs]def sub_forms_by_domain(form):
"Return a list of forms each with an integration domain"
if not isinstance(form, Form):
error("Unable to convert object to a UFL form: %s" % ufl_err_str(form))
return [Form(form.integrals_by_domain(domain)) for domain in form.ufl_domains()]

[docs]def as_form(form):
"Convert to form if not a form, otherwise return form."
if not isinstance(form, Form):
error("Unable to convert object to a UFL form: %s" % ufl_err_str(form))
return form

[docs]def replace_integral_domains(form, common_domain):  # TODO: Move elsewhere
"""Given a form and a domain, assign a common integration domain to
all integrals.

Does not modify the input form (``Form`` should always be
immutable).  This is to support ill formed forms with no domain
specified, sometimes occurring in pydolfin, e.g. assemble(1*dx,
mesh=mesh).

"""
domains = form.ufl_domains()
if common_domain is not None:
gdim = common_domain.geometric_dimension()
tdim = common_domain.topological_dimension()
if not all((gdim == domain.geometric_dimension() and
tdim == domain.topological_dimension())
for domain in domains):
error("Common domain does not share dimensions with form domains.")

reconstruct = False
integrals = []
for itg in form.integrals():
domain = itg.ufl_domain()
if domain != common_domain:
itg = itg.reconstruct(domain=common_domain)
reconstruct = True
integrals.append(itg)
if reconstruct:
form = Form(integrals)
return form
```