form.py

"""The Form class."""
# Copyright (C) 2008-2016 Martin Sandve Alnæs
#
# This file is part of UFL (https://www.fenicsproject.org)
#
# SPDX-License-Identifier:    LGPL-3.0-or-later
#
# Modified by Anders Logg, 2009-2011.
# Modified by Massimiliano Leoni, 2016.
# Modified by Cecile Daversin-Catty, 2018.
# Modified by Nacime Bouziani, 2020.
# Modified by Jørgen S. Dokken 2023.

import numbers
import warnings
from collections import defaultdict
from itertools import chain

from ufl.checks import is_scalar_constant_expression
from ufl.constant import Constant
from ufl.constantvalue import Zero
from ufl.core.expr import Expr, ufl_err_str
from ufl.core.ufl_type import UFLType, ufl_type
from ufl.domain import extract_unique_domain, sort_domains
from ufl.equation import Equation
from ufl.integral import Integral
from ufl.utils.counted import Counted
from ufl.utils.sorting import sorted_by_count

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

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


def _sorted_integrals(integrals):
    """Sort integrals for a stable signature computation.

    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:
            raise ValueError(
                "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].append(integral)

    all_integrals = []

    def keyfunc(item):
        if isinstance(item, numbers.Integral):
            sid_int = item
        else:
            # As subdomain ids can be either int or tuples, we need to compare them
            sid_int = tuple(-1 if i == "otherwise" else i for i in item)
        return (type(item).__name__, sid_int)

    # 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=keyfunc):
                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 accumulate_integrands_with_same_metadata 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


@ufl_type()
class BaseForm(object, metaclass=UFLType):
    """Description of an object containing arguments."""

    # Slots is kept empty to enable multiple inheritance with other
    # classes
    __slots__ = ()
    _ufl_is_abstract_ = True
    _ufl_required_methods_ = ("_analyze_form_arguments", "_analyze_domains", "ufl_domains")

    def __init__(self):
        """Initialise."""
        # Internal variables for caching form argument/coefficient data
        self._arguments = None
        self._coefficients = None

    # --- Accessor interface ---
    def arguments(self):
        """Return all ``Argument`` objects found in form."""
        if self._arguments is None:
            self._analyze_form_arguments()
        return self._arguments

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

    def geometric_quantities(self):
        """Return all ``GeometricQuantity`` objects found in form."""
        if self._geometric_quantities is None:
            self._analyze_form_arguments()
        return self._geometric_quantities

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

        Fails if multiple domains are found.
        """
        try:
            (domain,) = set(self.ufl_domains())
        except ValueError:
            raise ValueError("%s must have exactly one domain." % type(self).__name__)
        # Return the one and only domain
        return domain

    def empty(self):
        """Returns whether the BaseForm has no components."""
        return False

    # --- Operator implementations ---

    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 __radd__(self, other):
        """Add."""
        # Ordering of form additions make no difference
        return self.__add__(other)

    def __add__(self, other):
        """Add."""
        if isinstance(other, numbers.Number) and other == 0:
            # Allow adding 0 or 0.0 as a no-op, needed for sum([a,b])
            return self
        elif isinstance(other, Zero):
            # Allow adding ufl Zero as a no-op, needed for sum([a,b])
            return self

        elif isinstance(other, ZeroBaseForm):
            # Simplify addition with ZeroBaseForm
            return self

        # For `ZeroBaseForm(...) + B` with B a BaseForm.
        # We could overwrite ZeroBaseForm.__add__ but that implies
        # duplicating cases with `0` and `ufl.Zero`.
        elif isinstance(self, ZeroBaseForm):
            # Simplify addition with ZeroBaseForm
            return other

        elif isinstance(other, BaseForm):
            # Add integrals from both forms
            return FormSum((self, 1), (other, 1))

        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.
        """
        if isinstance(self, ZeroBaseForm):
            # `-` doesn't change anything for ZeroBaseForm.
            # This also facilitates simplifying FormSum containing ZeroBaseForm objects.
            return self
        return FormSum((self, -1))

    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 FormSum((self, scalar))
        return NotImplemented

    def __mul__(self, coefficient):
        """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 __ne__(self, other):
        """Immediately evaluate the != operator (as opposed to the == operator)."""
        return not self.equals(other)

    def __call__(self, x):
        """Take the action of this form on ``x``."""
        from ufl.formoperators import action

        return action(self, x)

    def _ufl_compute_hash_(self):
        """Compute the hash."""
        # Ensure compatibility with MultiFunction
        # `hash(self)` will call the `__hash__` method of the subclass.
        return hash(self)

    def _ufl_expr_reconstruct_(self, *operands):
        """Return a new object of the same type with new operands."""
        return type(self)(*operands)

    __matmul__ = __mul__


@ufl_type()
class Form(BaseForm):
    """Description of a weak form consisting of a sum of integrals over subdomains."""

    __slots__ = (
        "_arguments",
        "_base_form_operators",
        # --- Dict that external frameworks can place framework-specific
        #     data in to be carried with the form
        #     Never use this internally in ufl!
        "_cache",
        "_coefficient_numbering",
        "_coefficients",
        "_constant_numbering",
        "_constants",
        "_domain_numbering",
        "_geometric_quantities",
        "_hash",
        # --- 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",
        "_signature",
        "_subdomain_data",
        "_terminal_numbering",
    )

    def __init__(self, integrals):
        """Initialise."""
        BaseForm.__init__(self)
        # Basic input checking (further compatibilty analysis happens
        # later)
        if not all(isinstance(itg, Integral) for itg in integrals):
            raise ValueError("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._coefficients = None
        self._coefficient_numbering = None
        self._constant_numbering = None
        self._terminal_numbering = None

        # Internal variables for caching base form operator data
        self._base_form_operators = None

        from ufl.algorithms.analysis import extract_constants

        self._constants = extract_constants(self)

        # 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 ---

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

    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
        )

    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)

    def empty(self):
        """Returns whether the form has no integrals."""
        return len(self.integrals()) == 0

    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

    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()

    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:
            raise ValueError(
                "Expecting all domains and functions in a form "
                f"to share geometric dimension, got {tuple(sorted(gdims))}"
            )
        return gdims[0]

    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

    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

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

    def base_form_operators(self):
        """Return all ``BaseFormOperator`` objects found in form."""
        if self._base_form_operators is None:
            self._analyze_base_form_operators()
        return self._base_form_operators

    def coefficient_numbering(self):
        """Return a contiguous numbering of coefficients in a mapping ``{coefficient:number}``."""
        # cyclic import
        from ufl.coefficient import Coefficient

        if self._coefficient_numbering is None:
            self._coefficient_numbering = {
                expr: num
                for expr, num in self.terminal_numbering().items()
                if isinstance(expr, Coefficient)
            }
        return self._coefficient_numbering

    def constants(self):
        """Get constants."""
        return self._constants

    def constant_numbering(self):
        """Return a contiguous numbering of constants in a mapping ``{constant:number}``."""
        if self._constant_numbering is None:
            self._constant_numbering = {
                expr: num
                for expr, num in self.terminal_numbering().items()
                if isinstance(expr, Constant)
            }
        return self._constant_numbering

    def terminal_numbering(self):
        """Return a contiguous numbering for all counted objects in the form.

        The returned object is mapping from terminal to its number (an integer).

        The numbering is computed per type so :class:`Coefficient`s,
        :class:`Constant`s, etc will each be numbered from zero.
        """
        # cyclic import
        from ufl.algorithms.analysis import extract_type

        if self._terminal_numbering is None:
            exprs_by_type = defaultdict(set)
            for counted_expr in extract_type(self, Counted):
                exprs_by_type[counted_expr._counted_class].add(counted_expr)

            numbering = {}
            for exprs in exprs_by_type.values():
                for i, expr in enumerate(sorted_by_count(exprs)):
                    numbering[expr] = i
            self._terminal_numbering = numbering
        return self._terminal_numbering

    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."""
        if self._hash is None:
            self._hash = hash(tuple(hash(itg) for itg in self.integrals()))
        return self._hash

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

    def equals(self, other):
        """Evaluate ``bool(lhs_form == rhs_form)``."""
        if type(other) is not 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))

    def __radd__(self, other):
        """Add."""
        # Ordering of form additions make no difference
        return self.__add__(other)

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

        if isinstance(other, ZeroBaseForm):
            # Simplify addition with ZeroBaseForm
            return self

        elif isinstance(other, BaseForm):
            # Create form sum if form is of other type
            return FormSum((self, 1), (other, 1))

        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)
          a = g*inner(grad(u), grad(v))*dx
          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):
                raise ValueError(f"Need {len(arguments)} arguments to form(), got {len(args)}.")
            repdict.update(zip(arguments, args))

        coefficients = kwargs.pop("coefficients", None)
        if kwargs:
            raise ValueError(f"Unknown kwargs {list(kwargs)}")

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

            return replace(self, repdict)
        else:
            return self

    __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(map(str, 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(map(repr, self.integrals()))
        r = "Form([" + itgs + "])"
        return r

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

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

        # Collect integration domains.
        self._integration_domains = sort_domains(
            join_domains([itg.ufl_domain() for itg in self._integrals])
        )
        # Collect domains in integrands.
        domains_in_integrands = set()
        for o in chain(
            self.arguments(), self.coefficients(), self.constants(), self.geometric_quantities()
        ):
            domain = extract_unique_domain(o, expand_mixed_mesh=False)
            domains_in_integrands.update(domain.meshes)
        domains_in_integrands -= set(self._integration_domains)
        all_domains = self._integration_domains + sort_domains(join_domains(domains_in_integrands))
        # Let problem solving environments access all domains via
        # self._domain_numbering.keys() (wrapped in extract_domains()).
        self._domain_numbering = {d: i for i, d in enumerate(all_domains)}

    def _analyze_subdomain_data(self):
        """Analyze subdomain data."""
        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
            if subdomain_data[domain].get(it) is None:
                subdomain_data[domain][it] = [sd]
            else:
                subdomain_data[domain][it].append(sd)
        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_terminals_with_domain

        arguments, coefficients, geometric_quantities = extract_terminals_with_domain(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._geometric_quantities = geometric_quantities  # sorted by (type, domain)

    def _analyze_base_form_operators(self):
        """Analyze which BaseFormOperator objects can be found in the form."""
        from ufl.algorithms.analysis import extract_base_form_operators

        base_form_ops = extract_base_form_operators(self)
        self._base_form_operators = tuple(sorted(base_form_ops, key=lambda x: x.count()))

    def _compute_renumbering(self):
        """Compute renumbering."""
        dn = self.domain_numbering()
        tn = self.terminal_numbering()
        renumbering = {}
        renumbering.update(dn)
        renumbering.update(tn)
        return renumbering

    def _compute_signature(self):
        """Compute signature."""
        from ufl.algorithms.signature import compute_form_signature

        self._signature = compute_form_signature(self, self._compute_renumbering())


def as_form(form):
    """Convert to form if not a form, otherwise return form."""
    if not isinstance(form, BaseForm) and form != 0:
        raise ValueError(f"Unable to convert object to a UFL form: {ufl_err_str(form)}")
    return form


@ufl_type()
class FormSum(BaseForm):
    """Form sum.

    Description of a weighted sum of variational forms and form-like objects
    components is the list of Forms to be summed
    arg_weights is a list of tuples of component index and weight
    """

    __slots__ = (
        "_arguments",
        "_coefficients",
        "_components",
        "_domain_numbering",
        "_domains",
        "_hash",
        "_weights",
        "ufl_operands",
    )
    _ufl_required_methods_ = "_analyze_form_arguments"

    def __new__(cls, *args, **kwargs):
        """Create a new FormSum."""
        # All the components are `ZeroBaseForm`
        if all(component == 0 for component, _ in args):
            # Assume that the arguments of all the components have
            # consistent with each other  and select the first one to
            # define the arguments of `ZeroBaseForm`.
            # This might not always be true but `ZeroBaseForm`'s
            # arguments are not checked anywhere because we can't
            # reliably always infer them.
            ((arg, _), *_) = args
            arguments = arg.arguments()
            return ZeroBaseForm(arguments)

        return super(FormSum, cls).__new__(cls)

    def __init__(self, *components):
        """Initialise."""
        BaseForm.__init__(self)

        # Remove `ZeroBaseForm` components
        filtered_components = [(component, w) for component, w in components if component != 0]

        weights = []
        full_components = []
        for component, w in filtered_components:
            if isinstance(component, FormSum):
                full_components.extend(component.components())
                weights.extend([w * wc for wc in component.weights()])
            else:
                full_components.append(component)
                weights.append(w)

        self._arguments = None
        self._coefficients = None
        self._domains = None
        self._domain_numbering = None
        self._hash = None
        self._weights = weights
        self._components = full_components
        self._sum_variational_components()
        self.ufl_operands = self._components

    def components(self):
        """Get components."""
        return self._components

    def weights(self):
        """Get weights."""
        return self._weights

    def _sum_variational_components(self):
        """Sum variational components."""
        var_forms = None
        other_components = []
        new_weights = []
        for i, component in enumerate(self._components):
            if isinstance(component, Form):
                if var_forms:
                    var_forms = var_forms + (self._weights[i] * component)
                else:
                    var_forms = self._weights[i] * component
            else:
                other_components.append(component)
                new_weights.append(self._weights[i])
        if var_forms:
            other_components.insert(0, var_forms)
            new_weights.insert(0, 1)
        self._components = other_components
        self._weights = new_weights

    def _analyze_form_arguments(self):
        """Return all ``Argument`` objects found in form."""
        arguments = []
        coefficients = []
        for component in self._components:
            arguments.extend(component.arguments())
            coefficients.extend(component.coefficients())
        # 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()))

    def _analyze_domains(self):
        """Analyze which domains can be found in FormSum."""
        from ufl.domain import join_domains, sort_domains

        # Collect unique domains
        self._domains = sort_domains(
            join_domains(chain.from_iterable(c.ufl_domains() for c in self.components()))
        )

    def ufl_domains(self):
        """Return all domains found in the base form."""
        if self._domains is None:
            self._analyze_domains()
        return self._domains

    def __hash__(self):
        """Hash."""
        if self._hash is None:
            self._hash = hash(
                tuple((hash(c), hash(w)) for c, w in zip(self.components(), self.weights()))
            )
        return self._hash

    def equals(self, other):
        """Evaluate ``bool(lhs_form == rhs_form)``."""
        if type(other) is not FormSum:
            return False
        if self is other:
            return True
        return (
            len(self.components()) == len(other.components())
            and all(a == b for a, b in zip(self.components(), other.components()))
            and all(a == b for a, b in zip(self.weights(), other.weights()))
        )

    def empty(self):
        """Returns whether the FormSum has no components."""
        return len(self.components()) == 0

    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(f"{w}*{c}" for c, w in zip(self.components(), self.weights()))
        return s or "<empty FormSum>"

    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(f"{w!r}*{c!r}" for c, w in zip(self.components(), self.weights()))
        r = "FormSum([" + itgs + "])"
        return r


@ufl_type()
class ZeroBaseForm(BaseForm):
    """Description of a zero base form.

    ZeroBaseForm is idempotent with respect to assembly and is mostly
    used for sake of simplifying base-form expressions.
    """

    __slots__ = (
        "_arguments",
        "_coefficients",
        "_domains",
        "_hash",
        # Pyadjoint compatibility
        "form",
        "ufl_operands",
    )

    def __init__(self, arguments):
        """Initialise."""
        BaseForm.__init__(self)
        self._arguments = arguments
        self.ufl_operands = arguments
        self._hash = None
        self._domains = None
        self.form = None

    def _analyze_form_arguments(self):
        """Analyze form arguments."""
        # `self._arguments` is already set in `BaseForm.__init__`
        self._coefficients = ()

    def _analyze_domains(self):
        """Analyze which domains can be found in ZeroBaseForm."""
        from ufl.domain import join_domains, sort_domains

        # Collect unique domains
        self._domains = sort_domains(
            join_domains(chain.from_iterable(e.ufl_domains() for e in self.ufl_operands))
        )

    def ufl_domains(self):
        """Return all domains found in the base form."""
        if self._domains is None:
            self._analyze_domains()
        return self._domains

    def __ne__(self, other):
        """Overwrite BaseForm.__neq__ which relies on `equals`."""
        return not self == other

    def __eq__(self, other):
        """Check equality."""
        if type(other) is ZeroBaseForm:
            if self is other:
                return True
            return self._arguments == other._arguments
        elif isinstance(other, (int, float)):
            return other == 0
        else:
            return False

    def __str__(self):
        """Format as a string."""
        return "ZeroBaseForm(%s)" % (", ".join(str(arg) for arg in self._arguments))

    def __repr__(self):
        """Representation."""
        return "ZeroBaseForm(%s)" % (", ".join(repr(arg) for arg in self._arguments))

    def __hash__(self):
        """Hash."""
        if self._hash is None:
            self._hash = hash(("ZeroBaseForm", hash(self._arguments)))
        return self._hash