"""This module defines the UFL finite element classes."""
# 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 Kristian B. Oelgaard
# Modified by Marie E. Rognes 2010, 2012
# Modified by Massimiliano Leoni, 2016
# Modified by Matthew Scroggs, 2023
from __future__ import annotations
import abc as _abc
import typing as _typing
import numpy as np
from ufl.cell import Cell as _Cell
from ufl.pullback import AbstractPullback as _AbstractPullback
from ufl.pullback import IdentityPullback as _IdentityPullback
from ufl.pullback import MixedPullback as _MixedPullback
from ufl.pullback import SymmetricPullback as _SymmetricPullback
from ufl.sobolevspace import SobolevSpace as _SobolevSpace
from ufl.utils.sequences import product
__all_classes__ = ["AbstractFiniteElement", "FiniteElement", "MixedElement", "SymmetricElement"]
[docs]class AbstractFiniteElement(_abc.ABC):
"""Base class for all finite elements.
To make your element library compatible with UFL, you should make a
subclass of AbstractFiniteElement and provide implementions of all
the abstract methods and properties. All methods and properties that
are not marked as abstract are implemented here and should not need
to be overwritten in your subclass.
An example of how the methods in your subclass could be implemented
can be found in Basix; see
https://github.com/FEniCS/basix/blob/main/python/basix/ufl.py
"""
@_abc.abstractmethod
def __repr__(self) -> str:
"""Format as string for evaluation as Python object."""
@_abc.abstractmethod
def __str__(self) -> str:
"""Format as string for nice printing."""
@_abc.abstractmethod
def __hash__(self) -> int:
"""Return a hash."""
@_abc.abstractmethod
def __eq__(self, other: AbstractFiniteElement) -> bool:
"""Check if this element is equal to another element."""
@_abc.abstractproperty
def sobolev_space(self) -> _SobolevSpace:
"""Return the underlying Sobolev space."""
@_abc.abstractproperty
def pullback(self) -> _AbstractPullback:
"""Return the pullback for this element."""
@_abc.abstractproperty
def embedded_superdegree(self) -> _typing.Union[int, None]:
"""Degree of the minimum degree Lagrange space that spans this element.
This returns the degree of the lowest degree Lagrange space such
that the polynomial space of the Lagrange space is a superspace
of this element's polynomial space. If this element contains
basis functions that are not in any Lagrange space, this
function should return None.
Note that on a simplex cells, the polynomial space of Lagrange
space is a complete polynomial space, but on other cells this is
not true. For example, on quadrilateral cells, the degree 1
Lagrange space includes the degree 2 polynomial xy.
"""
@_abc.abstractproperty
def embedded_subdegree(self) -> int:
"""Degree of the maximum degree Lagrange space that is spanned by this element.
This returns the degree of the highest degree Lagrange space
such that the polynomial space of the Lagrange space is a
subspace of this element's polynomial space. If this element's
polynomial space does not include the constant function, this
function should return -1.
Note that on a simplex cells, the polynomial space of Lagrange
space is a complete polynomial space, but on other cells this is
not true. For example, on quadrilateral cells, the degree 1
Lagrange space includes the degree 2 polynomial xy.
"""
@_abc.abstractproperty
def cell(self) -> _Cell:
"""Return the cell of the finite element."""
@_abc.abstractproperty
def reference_value_shape(self) -> _typing.Tuple[int, ...]:
"""Return the shape of the value space on the reference cell."""
@_abc.abstractproperty
def sub_elements(self) -> _typing.List:
"""Return list of sub-elements.
This function does not recurse: ie it does not extract the sub-elements
of sub-elements.
"""
def __ne__(self, other: AbstractFiniteElement) -> bool:
"""Check if this element is different to another element."""
return not self.__eq__(other)
[docs] def is_cellwise_constant(self) -> bool:
"""Check whether this element is spatially constant over each cell."""
return self.embedded_superdegree == 0
def _ufl_hash_data_(self) -> str:
"""Return UFL hash data."""
return repr(self)
def _ufl_signature_data_(self) -> str:
"""Return UFL signature data."""
return repr(self)
@property
def components(self) -> _typing.Dict[_typing.Tuple[int, ...], int]:
"""Get the numbering of the components of the element.
Returns:
A map from the components of the values on a physical cell (eg (0, 1))
to flat component numbers on the reference cell (eg 1)
"""
if isinstance(self.pullback, _SymmetricPullback):
return self.pullback._symmetry
if len(self.sub_elements) == 0:
return {(): 0}
components = {}
offset = 0
c_offset = 0
for e in self.sub_elements:
for i, j in enumerate(np.ndindex(e.value_shape)):
components[(offset + i,)] = c_offset + e.components[j]
c_offset += max(e.components.values()) + 1
offset += e.value_size
return components
@property
def reference_value_size(self) -> int:
"""Return the integer product of the reference value shape."""
return product(self.reference_value_shape)
@property
def num_sub_elements(self) -> int:
"""Return number of sub-elements.
This function does not recurse: ie it does not count the sub-elements of
sub-elements.
"""
return len(self.sub_elements)
[docs]class FiniteElement(AbstractFiniteElement):
"""A directly defined finite element."""
__slots__ = (
"_repr",
"_str",
"_family",
"_cell",
"_degree",
"_reference_value_shape",
"_pullback",
"_sobolev_space",
"_sub_elements",
"_subdegree",
)
def __init__(
self,
family: str,
cell: _Cell,
degree: int,
reference_value_shape: _typing.Tuple[int, ...],
pullback: _AbstractPullback,
sobolev_space: _SobolevSpace,
sub_elements=[],
_repr: _typing.Optional[str] = None,
_str: _typing.Optional[str] = None,
subdegree: _typing.Optional[int] = None,
):
"""Initialise a finite element.
This class should only be used for testing
Args:
family: The family name of the element
cell: The cell on which the element is defined
degree: The polynomial degree of the element
reference_value_shape: The reference value shape of the element
pullback: The pullback to use
sobolev_space: The Sobolev space containing this element
sub_elements: Sub elements of this element
_repr: A string representation of this elements
_str: A string for printing
subdegree: The embedded subdegree of this element
"""
if subdegree is None:
self._subdegree = degree
else:
self._subdegree = subdegree
if _repr is None:
if len(sub_elements) > 0:
self._repr = (
f'ufl.finiteelement.FiniteElement("{family}", {cell}, {degree}, '
f"{reference_value_shape}, {pullback}, {sobolev_space}, {sub_elements!r})"
)
else:
self._repr = (
f'ufl.finiteelement.FiniteElement("{family}", {cell}, {degree}, '
f"{reference_value_shape}, {pullback}, {sobolev_space})"
)
else:
self._repr = _repr
if _str is None:
self._str = f"<{family}{degree} on a {cell}>"
else:
self._str = _str
self._family = family
self._cell = cell
self._degree = degree
self._reference_value_shape = reference_value_shape
self._pullback = pullback
self._sobolev_space = sobolev_space
self._sub_elements = sub_elements
def __repr__(self) -> str:
"""Format as string for evaluation as Python object."""
return self._repr
def __str__(self) -> str:
"""Format as string for nice printing."""
return self._str
def __hash__(self) -> int:
"""Return a hash."""
return hash(f"{self!r}")
def __eq__(self, other) -> bool:
"""Check if this element is equal to another element."""
return type(self) is type(other) and repr(self) == repr(other)
@property
def sobolev_space(self) -> _SobolevSpace:
"""Return the underlying Sobolev space."""
return self._sobolev_space
@property
def pullback(self) -> _AbstractPullback:
"""Return the pullback for this element."""
return self._pullback
@property
def embedded_superdegree(self) -> _typing.Union[int, None]:
"""Degree of the minimum degree Lagrange space that spans this element.
This returns the degree of the lowest degree Lagrange space such
that the polynomial space of the Lagrange space is a superspace
of this element's polynomial space. If this element contains
basis functions that are not in any Lagrange space, this
function should return None.
Note that on a simplex cells, the polynomial space of Lagrange
space is a complete polynomial space, but on other cells this is
not true. For example, on quadrilateral cells, the degree 1
Lagrange space includes the degree 2 polynomial xy.
"""
return self._degree
@property
def embedded_subdegree(self) -> int:
"""Degree of the maximum degree Lagrange space that is spanned by this element.
This returns the degree of the highest degree Lagrange space
such that the polynomial space of the Lagrange space is a
subspace of this element's polynomial space. If this element's
polynomial space does not include the constant function, this
function should return -1.
Note that on a simplex cells, the polynomial space of Lagrange
space is a complete polynomial space, but on other cells this is
not true. For example, on quadrilateral cells, the degree 1
Lagrange space includes the degree 2 polynomial xy.
"""
return self._subdegree
@property
def cell(self) -> _Cell:
"""Return the cell of the finite element."""
return self._cell
@property
def reference_value_shape(self) -> _typing.Tuple[int, ...]:
"""Return the shape of the value space on the reference cell."""
return self._reference_value_shape
@property
def sub_elements(self) -> _typing.List:
"""Return list of sub-elements.
This function does not recurse: ie it does not extract the
sub-elements of sub-elements.
"""
return self._sub_elements
[docs]class SymmetricElement(FiniteElement):
"""A symmetric finite element."""
def __init__(
self,
symmetry: _typing.Dict[_typing.Tuple[int, ...], int],
sub_elements: _typing.List[AbstractFiniteElement],
):
"""Initialise a symmetric element.
This class should only be used for testing
Args:
symmetry: Map from physical components to reference components
sub_elements: Sub-elements of this element
"""
self._sub_elements = sub_elements
pullback = _SymmetricPullback(self, symmetry)
reference_value_shape = (sum(e.reference_value_size for e in sub_elements),)
degree = max(e.embedded_superdegree for e in sub_elements)
cell = sub_elements[0].cell
for e in sub_elements:
if e.cell != cell:
raise ValueError("All sub-elements must be defined on the same cell")
sobolev_space = max(e.sobolev_space for e in sub_elements)
super().__init__(
"Symmetric element",
cell,
degree,
reference_value_shape,
pullback,
sobolev_space,
sub_elements=sub_elements,
_repr=(f"ufl.finiteelement.SymmetricElement({symmetry!r}, {sub_elements!r})"),
_str=f"<symmetric element on a {cell}>",
)
[docs]class MixedElement(FiniteElement):
"""A mixed element."""
def __init__(self, sub_elements):
"""Initialise a mixed element.
This class should only be used for testing
Args:
sub_elements: Sub-elements of this element
"""
sub_elements = [MixedElement(e) if isinstance(e, list) else e for e in sub_elements]
cell = sub_elements[0].cell
for e in sub_elements:
assert e.cell == cell
degree = max(e.embedded_superdegree for e in sub_elements)
reference_value_shape = (sum(e.reference_value_size for e in sub_elements),)
if all(isinstance(e.pullback, _IdentityPullback) for e in sub_elements):
pullback = _IdentityPullback()
else:
pullback = _MixedPullback(self)
sobolev_space = max(e.sobolev_space for e in sub_elements)
super().__init__(
"Mixed element",
cell,
degree,
reference_value_shape,
pullback,
sobolev_space,
sub_elements=sub_elements,
_repr=f"ufl.finiteelement.MixedElement({sub_elements!r})",
_str=f"<MixedElement with {len(sub_elements)} sub-element(s)>",
)