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forms.py
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787 lines (667 loc) · 28.3 KB
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# Copyright (C) 2017-2026 Chris N. Richardson, Garth N. Wells,
# Michal Habera, Jørgen S. Dokken, Jack S. Hale and Jose Fernandez
#
# This file is part of DOLFINx (https://www.fenicsproject.org)
#
# SPDX-License-Identifier: LGPL-3.0-or-later
"""Finite element forms."""
from __future__ import annotations
import collections
import types
import typing
from collections.abc import Sequence
from dataclasses import dataclass
from mpi4py import MPI
import numpy as np
import numpy.typing as npt
import ffcx
import ufl
from dolfinx import cpp as _cpp
from dolfinx import default_scalar_type, jit
from dolfinx.fem import IntegralType
from dolfinx.fem.function import Constant, Function, FunctionSpace
if typing.TYPE_CHECKING:
# import dolfinx.mesh just when doing type checking to avoid
# circular import
from dolfinx.mesh import EntityMap as _EntityMap
from dolfinx.mesh import Mesh, MeshTags
class Form:
"""A finite element form."""
_cpp_object: (
_cpp.fem.Form_complex64
| _cpp.fem.Form_complex128
| _cpp.fem.Form_float32
| _cpp.fem.Form_float64
)
_code: str | list[str] | None
def __init__(
self,
form: _cpp.fem.Form_complex64
| _cpp.fem.Form_complex128
| _cpp.fem.Form_float32
| _cpp.fem.Form_float64,
ufcx_form=None,
code: str | list[str] | None = None,
module: types.ModuleType | list[types.ModuleType] | None = None,
):
"""Initialize a finite element form.
Note:
Forms should normally be constructed using :func:`form` and
not using this class initialiser. This class is combined
with different base classes that depend on the scalar type
used in the Form.
Args:
form: Compiled form object.
ufcx_form: UFCx form.
code: Form C++ code.
module: CFFI module.
"""
self._code = code
self._ufcx_form = ufcx_form
self._cpp_object = form
self._module = module
@property
def ufcx_form(self):
"""The compiled ufcx_form object."""
return self._ufcx_form
@property
def code(self) -> str | list[str] | None:
"""C code strings."""
return self._code
@property
def module(self) -> types.ModuleType | list[types.ModuleType] | None:
"""The CFFI module."""
return self._module
@property
def rank(self) -> int:
"""Rank of this form."""
return self._cpp_object.rank
@property
def function_spaces(self) -> list[FunctionSpace]:
"""Function spaces on which this form is defined."""
return self._cpp_object.function_spaces
@property
def dtype(self) -> np.dtype:
"""Scalar type of this form."""
return np.dtype(self._cpp_object.dtype)
@property
def mesh(self) -> _cpp.mesh.Mesh_float32 | _cpp.mesh.Mesh_float64:
"""Mesh on which this form is defined."""
return self._cpp_object.mesh
@property
def integral_types(self):
"""Integral types in the form."""
return self._cpp_object.integral_types
def num_integrals(self, integral_type: IntegralType, kernel_index: int) -> int:
"""Number of integrals of a given type for a specific cell type.
Args:
integral_type: The type of integral to count.
kernel_index: In the case of mixed topology, we have a kernel
per cell type. For single-cell type meshes, this is zero.
"""
return self._cpp_object.num_integrals(integral_type, kernel_index)
def get_integration_domains(
integral_type: IntegralType,
subdomain: MeshTags | list[tuple[int, np.ndarray]] | None,
subdomain_ids: list[int],
) -> list[tuple[int, np.ndarray]]:
"""Get integration domains from subdomain data.
The subdomain data is a MeshTags object consisting of markers, or
``None``. If it is ``None``, we do not pack any integration
entities. Integration domains are defined as a list of tuples, where
each input ``subdomain_ids`` is mapped to an array of integration
entities, where an integration entity for a cell integral is the
list of cells. For an exterior facet integral each integration
entity is a tuple ``(cell_index, local_facet_index)``. For an
interior facet integral each integration entity is a tuple
``(cell_index0, local_facet_index0, cell_index1,
local_facet_index1)``. Where the first cell-facet pair is the
``'+'`` restriction, the second the ``'-'`` restriction.
Args:
integral_type: The type of integral to pack integration
entities for.
subdomain: A MeshTag with markers or manually specified
integration domains.
subdomain_ids: List of ids to integrate over.
Returns:
A list of entities to integrate over. For cell integrals, this is a
list of cells. For exterior facet integrals, this is a list of
(cell, local_facet) pairs. For interior facet integrals, this is a
list of (cell0, local_facet0, cell1, local_facet1) tuples.
"""
if subdomain is None:
return []
else:
domains = []
if not isinstance(subdomain, list):
if integral_type in (IntegralType.exterior_facet, IntegralType.interior_facet):
tdim = subdomain.topology.dim
subdomain._cpp_object.topology.create_connectivity(tdim - 1, tdim)
subdomain._cpp_object.topology.create_connectivity(tdim, tdim - 1)
if integral_type is IntegralType.vertex:
tdim = subdomain.topology.dim
subdomain._cpp_object.topology.create_connectivity(0, tdim)
subdomain._cpp_object.topology.create_connectivity(tdim, 0)
if integral_type is IntegralType.ridge:
tdim = subdomain.topology.dim
subdomain._cpp_object.topology.create_connectivity(tdim - 2, tdim)
subdomain._cpp_object.topology.create_connectivity(tdim, tdim - 2)
# Special handling for exterior facets, compared to other
# one-sided entity integrals
if integral_type is IntegralType.exterior_facet:
exterior_facets = _cpp.mesh.exterior_facet_indices(subdomain.topology)
# Compute integration domains only for each subdomain id in
# the integrals. If a process has no integral entities,
# insert an empty array.
for id in subdomain_ids:
entities = subdomain.find(id)
if integral_type is IntegralType.exterior_facet:
# Compute intersection of tag an exterior facets
entities = np.intersect1d(entities, exterior_facets)
integration_entities = _cpp.fem.compute_integration_domains(
integral_type,
subdomain._cpp_object.topology,
entities,
)
domains.append((id, integration_entities))
return [(s[0], np.array(s[1])) for s in domains]
else:
return [(s[0], np.array(s[1])) for s in sorted(subdomain)]
def form_cpp_class(
dtype: npt.DTypeLike,
) -> (
_cpp.fem.Form_float32
| _cpp.fem.Form_float64
| _cpp.fem.Form_complex64
| _cpp.fem.Form_complex128
):
"""Wrapped C++ class of a variational form of a specific scalar type.
Args:
dtype: Scalar type of the required form class.
Returns:
Wrapped C++ form class of the requested type.
Note:
This function is for advanced usage, typically when writing
custom kernels using Numba or C.
"""
if np.issubdtype(dtype, np.float32):
return _cpp.fem.Form_float32
elif np.issubdtype(dtype, np.float64):
return _cpp.fem.Form_float64
elif np.issubdtype(dtype, np.complex64):
return _cpp.fem.Form_complex64
elif np.issubdtype(dtype, np.complex128):
return _cpp.fem.Form_complex128
else:
raise NotImplementedError(f"Type {dtype} not supported.")
_ufl_to_dolfinx_domain = {
"cell": IntegralType.cell,
"exterior_facet": IntegralType.exterior_facet,
"interior_facet": IntegralType.interior_facet,
"vertex": IntegralType.vertex,
"ridge": IntegralType.ridge,
}
def mixed_topology_form(
forms: Sequence[ufl.Form],
dtype: npt.DTypeLike = default_scalar_type,
form_compiler_options: dict | None = None,
jit_options: dict | None = None,
jit_comm: MPI.Intracomm | None = None,
entity_maps: Sequence[_EntityMap] | None = None,
):
"""Create a mixed-topology from from an array of Forms.
# FIXME: This function is a temporary hack for mixed-topology
meshes. # It is needed because UFL does not know about
mixed-topology meshes, # so we need to pass a list of forms for each
cell type.
Args:
forms: A list of UFL forms. Each form should be the same, just
defined on different cell types.
dtype: Scalar type to use for the compiled form.
form_compiler_options: See :func:`ffcx_jit <dolfinx.jit.ffcx_jit>`
jit_options: See :func:`ffcx_jit <dolfinx.jit.ffcx_jit>`.
jit_comm: MPI communicator used when compiling the form. If
``None``, then ``form.mesh.comm``.
entity_maps: If any trial functions, test functions, or
coefficients in the form are not defined over the same mesh
as the integration domain (the domain associated with the
measure), `entity_maps` must be supplied. For each mesh in
the form, there should be an entity map relating entities in
that mesh to the integration domain mesh.
Returns:
Compiled finite element Form.
"""
if form_compiler_options is None:
form_compiler_options = dict()
form_compiler_options["scalar_type"] = dtype
ftype = form_cpp_class(dtype)
# Extract subdomain data from UFL form
sd = next(iter(forms)).subdomain_data()
(domain,) = list(sd.keys()) # Assuming single domain
# Check that subdomain data for each integral type is the same
for data in sd.get(domain).values():
assert all([d is data[0] for d in data if d is not None])
mesh = domain.ufl_cargo()
if mesh is None:
raise RuntimeError("Expecting to find a Mesh in the form.")
comm = mesh.comm if jit_comm is None else jit_comm
ufcx_forms = []
modules = []
codes = []
for form in forms:
ufcx_form, module, code = jit.ffcx_jit(
comm,
form,
form_compiler_options=form_compiler_options,
jit_options=jit_options,
)
ufcx_forms.append(ufcx_form)
modules.append(module)
codes.append(code)
# In a mixed-topology mesh, each form has the same C++ function
# space, so we can extract it from any of them
V = [arg.ufl_function_space()._cpp_object for arg in form.arguments()]
# TODO coeffs, constants, subdomains, entity_maps
f = ftype(
[module.ffi.cast("uintptr_t", module.ffi.addressof(ufcx_form)) for ufcx_form in ufcx_forms],
V,
[],
[],
{},
[],
mesh,
)
return Form(f, ufcx_forms, codes, modules)
def form(
form: ufl.Form | Sequence[ufl.Form] | Sequence[Sequence[ufl.Form]],
dtype: npt.DTypeLike = default_scalar_type,
form_compiler_options: dict | None = None,
jit_options: dict | None = None,
jit_comm: MPI.Intracomm | None = None,
entity_maps: Sequence[_EntityMap] | None = None,
):
"""Create a Form or list of Forms.
Args:
form: A UFL form or iterable of UFL forms.
dtype: Scalar type to use for the compiled form.
form_compiler_options: See :func:`ffcx_jit <dolfinx.jit.ffcx_jit>`
jit_options: See :func:`ffcx_jit <dolfinx.jit.ffcx_jit>`.
jit_comm: MPI communicator used when compiling the form. If
`None`, then `form.mesh.comm`.
entity_maps: If any trial functions, test functions, or
coefficients in the form are not defined over the same mesh
as the integration domain (the domain associated with the
measure), `entity_maps` must be supplied. For each mesh in
the form, there should be an entity map relating entities in
that mesh to the integration domain mesh.
Returns:
Compiled finite element Form.
Note:
This function is responsible for the compilation of a UFL form
(using FFCx) and attaching coefficients and domains specific
data to the underlying C++ form. It dynamically create a
:class:`Form` instance with an appropriate base class for the
scalar type, e.g. :func:`_cpp.fem.Form_float64`.
"""
if form_compiler_options is None:
form_compiler_options = dict()
form_compiler_options["scalar_type"] = dtype
ftype = form_cpp_class(dtype)
def _form(form):
"""Compile a single UFL form."""
# Extract subdomain data from UFL form
sd = form.subdomain_data()
(domain,) = list(sd.keys()) # Assuming single domain
# Check that subdomain data for each integral type is the same
for data in sd.get(domain).values():
assert all([d is data[0] for d in data if d is not None])
msh = domain.ufl_cargo()
if msh is None:
raise RuntimeError("Expecting to find a Mesh in the form.")
comm = msh.comm if jit_comm is None else jit_comm
ufcx_form, module, code = jit.ffcx_jit(
comm, form, form_compiler_options=form_compiler_options, jit_options=jit_options
)
# For each argument in form extract its function space
V = [arg.ufl_function_space()._cpp_object for arg in form.arguments()]
part = form_compiler_options.get("part", "full")
if part == "diagonal":
V = [V[0]]
# Prepare coefficients data. For every coefficient in form take
# its C++ object.
original_coeffs = form.coefficients()
coeffs = [
original_coeffs[ufcx_form.original_coefficient_positions[i]]._cpp_object
for i in range(ufcx_form.num_coefficients)
]
constants = [c._cpp_object for c in form.constants()]
# Extract subdomain ids from ufcx_form
subdomain_ids = {type: [] for type in sd.get(domain).keys()}
integral_offsets = [ufcx_form.form_integral_offsets[i] for i in range(6)]
for i in range(len(integral_offsets) - 1):
integral_type = IntegralType(i)
for j in range(integral_offsets[i], integral_offsets[i + 1]):
subdomain_ids[integral_type.name].append(ufcx_form.form_integral_ids[j])
# Subdomain markers (possibly empty list for some integral types)
subdomains = {
_ufl_to_dolfinx_domain[key]: get_integration_domains(
_ufl_to_dolfinx_domain[key], subdomain_data[0], subdomain_ids[key]
)
for (key, subdomain_data) in sd.get(domain).items()
}
if entity_maps is None:
_entity_maps = []
else:
_entity_maps = [entity_map._cpp_object for entity_map in entity_maps]
f = ftype(
[module.ffi.cast("uintptr_t", module.ffi.addressof(ufcx_form))],
V,
coeffs,
constants,
subdomains,
_entity_maps,
msh,
)
return Form(f, ufcx_form, code, module)
def _zero_form(form):
"""Compile a single 'zero' UFL form.
I.e. a form with no integrals.
"""
V = [arg.ufl_function_space()._cpp_object for arg in form.arguments()]
assert len(V) > 0
msh = V[0].mesh
f = ftype(
spaces=V,
integrals={},
coefficients=[],
constants=[],
need_permutation_data=False,
entity_maps=[],
mesh=msh,
)
return Form(f)
def _create_form(form):
"""Recursively convert ufl.Forms to dolfinx.fem.Form.
Args:
form: UFL form or list of UFL forms to extract DOLFINx forms
from.
Returns:
A ``dolfinx.fem.Form`` or a list of ``dolfinx.fem.Form``.
"""
if isinstance(form, ufl.Form):
return _form(form)
elif isinstance(form, ufl.ZeroBaseForm):
return _zero_form(form)
elif isinstance(form, collections.abc.Iterable):
return list(map(lambda sub_form: _create_form(sub_form), form))
else:
return form
return _create_form(form)
def extract_function_spaces(
forms: Form | Sequence[Form] | Sequence[Sequence[Form]],
index: int = 0,
) -> FunctionSpace | list[None | FunctionSpace]:
"""Extract common function spaces from an array of forms.
If ``forms`` is a list of linear forms, this function returns of list
of the corresponding test function spaces. If ``forms`` is a 2D
array of bilinear forms, for ``index=0`` the list of common test
function spaces for each row is returned, and if ``index=1`` the
common trial function spaces for each column are returned.
Args:
forms: A list of forms or a 2D array of forms.
index: Index of the function space to extract. If ``index=0``,
the test function spaces are extracted, if ``index=1`` the
trial function spaces are extracted.
Returns:
List of function spaces.
"""
_forms = np.array(forms)
if _forms.ndim == 0:
form: Form = _forms.tolist()
return form.function_spaces[0] if form is not None else None
elif _forms.ndim == 1:
assert index == 0, "Expected index=0 for 1D array of forms"
for form in _forms:
if form is not None:
assert form.rank == 1, "Expected linear form"
return [form.function_spaces[0] if form is not None else None for form in forms] # type: ignore[union-attr]
elif _forms.ndim == 2:
assert index == 0 or index == 1, "Expected index=0 or index=1 for 2D array of forms"
extract_spaces = np.vectorize(
lambda form: form.function_spaces[index] if form is not None else None
)
V = extract_spaces(_forms)
def unique_spaces(V):
# Pick spaces from first column
V0 = V[:, 0]
# Iterate over each column
for col in range(1, V.shape[1]):
# Iterate over entry in column, updating if current
# space is None, or where both spaces are not None check
# that they are the same
for row in range(V.shape[0]):
if V0[row] is None and V[row, col] is not None:
V0[row] = V[row, col]
elif V0[row] is not None and V[row, col] is not None:
assert V0[row] is V[row, col], "Cannot extract unique function spaces"
return V0
if index == 0:
return list(unique_spaces(V))
elif index == 1:
return list(unique_spaces(V.transpose()))
raise RuntimeError("Unsupported array of forms")
@dataclass
class CompiledForm:
"""Compiled UFL form without associated DOLFINx data."""
ufl_form: ufl.Form # The original ufl form
ufcx_form: typing.Any # The compiled form
module: typing.Any # The module
code: str # The source code
dtype: npt.DTypeLike # data type used for the `ufcx_form`
def compile_form(
comm: MPI.Intracomm,
form: ufl.Form,
form_compiler_options: dict | None = {"scalar_type": default_scalar_type},
jit_options: dict | None = None,
) -> CompiledForm:
"""Compile UFL form without associated DOLFINx data.
Args:
comm: The MPI communicator used when compiling the form
form: The UFL form to compile
form_compiler_options: See :func:`ffcx_jit <dolfinx.jit.ffcx_jit>`
jit_options: See :func:`ffcx_jit <dolfinx.jit.ffcx_jit>`.
"""
p_ffcx = ffcx.get_options(form_compiler_options)
p_jit = jit.get_options(jit_options)
ufcx_form, module, code = jit.ffcx_jit(comm, form, p_ffcx, p_jit)
scalar_type: npt.DTypeLike = p_ffcx["scalar_type"] # type: ignore [assignment]
return CompiledForm(form, ufcx_form, module, code, scalar_type)
def form_cpp_creator(
dtype: npt.DTypeLike,
) -> (
_cpp.fem.Form_float32
| _cpp.fem.Form_float64
| _cpp.fem.Form_complex64
| _cpp.fem.Form_complex128
):
"""A wrapped C++ constructor for a form with a specified scalar type.
Args:
dtype: Scalar type of the required form class.
Returns:
Wrapped C++ form class of the requested type.
Note:
This function is for advanced usage, typically when writing
custom kernels using Numba or C.
"""
if np.issubdtype(dtype, np.float32):
return _cpp.fem.create_form_float32
elif np.issubdtype(dtype, np.float64):
return _cpp.fem.create_form_float64
elif np.issubdtype(dtype, np.complex64):
return _cpp.fem.create_form_complex64
elif np.issubdtype(dtype, np.complex128):
return _cpp.fem.create_form_complex128
else:
raise NotImplementedError(f"Type {dtype} not supported.")
def create_form(
form: CompiledForm,
V: list[FunctionSpace],
msh: Mesh,
subdomains: dict[IntegralType, list[tuple[int, np.ndarray]]],
coefficient_map: dict[ufl.Coefficient, Function],
constant_map: dict[ufl.Constant, Constant],
entity_maps: Sequence[_EntityMap] | None = None,
) -> Form:
"""Create a Form object from a data-independent compiled form.
Args:
form: Compiled ufl form,
V: List of function spaces associated with the form. Should
match the number of arguments in the form.
msh: Mesh to associate form with.
subdomains: A map from integral type to a list of pairs, where
each pair corresponds to a subdomain id and the set of of
integration entities to integrate over. Can be computed with
{py:func}`dolfinx.fem.compute_integration_domains`.
coefficient_map: Map from UFL coefficient to function with data.
constant_map: Map from UFL constant to constant with data.
to the integration domain ``msh``. The value of the map is
an array of integers, where the i-th entry is the entity in
the key mesh.
entity_maps: Entity maps to support cases where forms involve
sub-meshes.
Return:
A Form object.
"""
if entity_maps is None:
_entity_maps = []
else:
_entity_maps = [entity_map._cpp_object for entity_map in entity_maps]
_subdomain_data = subdomains.copy()
for _, idomain in _subdomain_data.items():
idomain.sort(key=lambda x: x[0])
# Extract all coefficients of the compiled form in correct order
coefficients = {}
original_coefficients = ufl.algorithms.extract_coefficients(form.ufl_form)
num_coefficients = form.ufcx_form.num_coefficients
for c in range(num_coefficients):
original_index = form.ufcx_form.original_coefficient_positions[c]
original_coeff = original_coefficients[original_index]
try:
coefficients[f"w{c}"] = coefficient_map[original_coeff]._cpp_object
except KeyError:
raise RuntimeError(f"Missing coefficient {original_coeff}")
# Extract all constants of the compiled form in correct order
# NOTE: Constants are not eliminated
original_constants = ufl.algorithms.analysis.extract_constants(form.ufl_form)
num_constants = form.ufcx_form.num_constants
if num_constants != len(original_constants):
raise RuntimeError(
f"Number of constants in compiled form ({num_constants})",
f"does not match the original form {len(original_constants)}",
)
constants = {}
for counter, constant in enumerate(original_constants):
try:
mapped_constant = constant_map[constant]
constants[f"c{counter}"] = mapped_constant._cpp_object
except KeyError:
raise RuntimeError(f"Missing constant {constant}")
ftype = form_cpp_creator(form.dtype)
f = ftype(
form.module.ffi.cast("uintptr_t", form.module.ffi.addressof(form.ufcx_form)),
[fs._cpp_object for fs in V],
coefficients,
constants,
_subdomain_data,
_entity_maps,
msh._cpp_object,
)
return Form(f, form.ufcx_form, form.code)
def _derive_univariate_residual(
F: ufl.Form,
u: Function,
du: ufl.Argument | None = None,
) -> ufl.Form:
if du is None:
du = ufl.TestFunction(u.function_space)
return ufl.derivative(F, u, du)
def _derive_block_residual(
F: ufl.Form,
u: Sequence[ufl.Form],
du: Sequence[ufl.Argument] | None = None,
) -> Sequence[ufl.Form]:
if du is None:
du = ufl.TestFunctions(ufl.MixedFunctionSpace(*(u_i.function_space for u_i in u)))
return ufl.extract_blocks(ufl.derivative(F, u, du))
def _derive_univariate_jacobian(
F: ufl.Form,
u: Function,
du: ufl.Argument | None = None,
) -> ufl.Form:
if du is None:
du = ufl.TrialFunction(u.function_space)
return ufl.derivative(F, u, du)
def _derive_block_jacobian(
F: Sequence[ufl.Form],
u: Sequence[ufl.Form],
du: Sequence[ufl.Argument] | None = None,
) -> Sequence[Sequence[ufl.Form]]:
if not isinstance(u, Sequence):
raise ValueError("When F is a sequence, u must be a sequence")
if du is None:
du = [ufl.TrialFunction(u_i.function_space) for u_i in u]
elif (not isinstance(du, Sequence) or not len(u) == len(du)):
raise ValueError(
"When F is a list of N forms, du must be a sequence "
"containing N functions"
)
return [[ufl.derivative(F_i, u_j, du_j) for u_j, du_j in zip(u, du)] for F_i in F]
def derivative_block(
F: ufl.Form | Sequence[ufl.Form],
u: Function | Sequence[Function],
du: ufl.Argument | Sequence[ufl.Argument] | None = None,
) -> ufl.Form | Sequence[ufl.Form] | Sequence[Sequence[ufl.Form]]:
"""Return the UFL derivative of a UFL rank zero form, or the UFL derivative
of a (list of) rank one form(s).
This is commonly used to derive a block residual from a functional, or to
derive a block Jacobian from a block residual.
Four cases are supported:
1. ``F`` is a rank-zero ``ufl.Form``, and ``u`` is a ``ufl.Function``.
Returns a ``ufl.Form`` representing the residual :math:`R =
\\frac{\\partial F}{\\partial u}[\\delta u]`. This is equivalent to
calling {py:func}`ufl.derivative` directly.
2. ``F`` is a rank-zero `ufl.Form``, and ``u`` is a list of ``ufl.Function``.
Returns a list of ``ufl.Form`` representing the block residual :math:`R`,
with :math:`R_i = \\frac{\\partial F}{\\partial u_i}[\\delta u_i]`, where
:math:`\\delta u_i` is a test subfunction of the mixed space defined by
``u``. This is equivalent to calling {py:func}`ufl.extract_blocks` on the
result from {py:func}`ufl.derivative`.
3. ``F`` is a rank-one `ufl.Form``, and ``u`` is a ``ufl.Function``.
Returns a ``ufl.Form`` representing the Jacobian :math:`J =
\\frac{\\partial F}{\\partial u}[\\delta u]`. This is equivalent to
calling {py:func}`ufl.derivative` directly.
4. ``F`` is a list of rank-one `ufl.Form``, and ``u`` is a list of
``ufl.Function``. Returns a list of lists representing the block Jacobian
:math:`J`, with :math:`J_{ij} = \\frac{\\partial F_i}{u_j}[\\delta u_j]`
using {py:func}`ufl.derivative` called component-wise.
Args:
F: UFL form(s) to be derived.
u: Function(s) with respect to the derivative is computed.
du: UFL argument(s) representing the direction of the derivative.
""" # noqa: D301
if isinstance(F, ufl.Form) and not F.arguments():
if isinstance(u, Function):
return _derive_univariate_residual(F, u, du)
elif isinstance(u, Sequence):
return _derive_block_residual(F, u, du)
else:
raise ValueError("u must be either a ufl.Function or a sequence of ufl.Function")
elif isinstance(F, ufl.Form) and len(F.arguments()) == 1:
return _derive_univariate_jacobian(F, u, du)
elif isinstance(F, Sequence):
return _derive_block_jacobian(F, u, du)
else:
raise ValueError(
"F must be either a UFL form (with rank zero or one), or a sequence of "
"rank-one UFL forms."
)