def _compute_integral_ir(form_data, form_index, prefix, element_numbers,
integral_names, parameters, visualise):
"""Compute intermediate represention for form integrals."""
_entity_types = {
"cell": "cell",
"exterior_facet": "facet",
"interior_facet": "facet",
"vertex": "vertex",
"custom": "cell"
}
# Iterate over groups of integrals
irs = []
for itg_data_index, itg_data in enumerate(form_data.integral_data):
logger.info("Computing IR for integral in integral group {}".format(
itg_data_index))
# Compute representation
entitytype = _entity_types[itg_data.integral_type]
cell = itg_data.domain.ufl_cell()
cellname = cell.cellname()
tdim = cell.topological_dimension()
assert all(tdim == itg.ufl_domain().topological_dimension()
for itg in itg_data.integrals)
ir = {
"integral_type": itg_data.integral_type,
"subdomain_id": itg_data.subdomain_id,
"rank": form_data.rank,
"geometric_dimension": form_data.geometric_dimension,
"topological_dimension": tdim,
"entitytype": entitytype,
"num_facets": cell.num_facets(),
"num_vertices": cell.num_vertices(),
"needs_oriented": form_needs_oriented_jacobian(form_data),
"enabled_coefficients": itg_data.enabled_coefficients,
"cell_shape": cellname
}
# Get element space dimensions
unique_elements = element_numbers.keys()
ir["element_dimensions"] = {
ufl_element: create_element(ufl_element).space_dimension()
for ufl_element in unique_elements
}
ir["element_ids"] = {
ufl_element: i
for i, ufl_element in enumerate(unique_elements)
}
# Create dimensions of primary indices, needed to reset the argument
# 'A' given to tabulate_tensor() by the assembler.
argument_dimensions = [
ir["element_dimensions"][ufl_element]
for ufl_element in form_data.argument_elements
]
# Compute shape of element tensor
if ir["integral_type"] == "interior_facet":
ir["tensor_shape"] = [2 * dim for dim in argument_dimensions]
else:
ir["tensor_shape"] = argument_dimensions
integral_type = itg_data.integral_type
cell = itg_data.domain.ufl_cell()
# Group integrands with the same quadrature rule
grouped_integrands = {}
for integral in itg_data.integrals:
md = integral.metadata() or {}
scheme = md["quadrature_rule"]
degree = md["quadrature_degree"]
if scheme == "custom":
points = md["quadrature_points"]
weights = md["quadrature_weights"]
elif scheme == "vertex":
# FIXME: Could this come from FIAT?
#
# The vertex scheme, i.e., averaging the function value in the
# vertices and multiplying with the simplex volume, is only of
# order 1 and inferior to other generic schemes in terms of
# error reduction. Equation systems generated with the vertex
# scheme have some properties that other schemes lack, e.g., the
# mass matrix is a simple diagonal matrix. This may be
# prescribed in certain cases.
if degree > 1:
warnings.warn(
"Explicitly selected vertex quadrature (degree 1), but requested degree is {}."
.format(degree))
if cellname == "tetrahedron":
points, weights = (numpy.array([[0.0, 0.0, 0.0],
[1.0, 0.0, 0.0],
[0.0, 1.0, 0.0],
[0.0, 0.0, 1.0]]),
numpy.array([
1.0 / 24.0, 1.0 / 24.0, 1.0 / 24.0,
1.0 / 24.0
]))
elif cellname == "triangle":
points, weights = (numpy.array([[0.0, 0.0], [1.0, 0.0],
[0.0, 1.0]]),
numpy.array(
[1.0 / 6.0, 1.0 / 6.0, 1.0 / 6.0]))
elif cellname == "interval":
# Trapezoidal rule
return (numpy.array([[0.0], [1.0]]),
numpy.array([1.0 / 2.0, 1.0 / 2.0]))
else:
(points, weights) = create_quadrature_points_and_weights(
integral_type, cell, degree, scheme)
points = numpy.asarray(points)
weights = numpy.asarray(weights)
rule = QuadratureRule(points, weights)
if rule not in grouped_integrands:
grouped_integrands[rule] = []
grouped_integrands[rule].append(integral.integrand())
sorted_integrals = {}
for rule, integrands in grouped_integrands.items():
integrands_summed = sorted_expr_sum(integrands)
integral_new = Integral(integrands_summed, itg_data.integral_type,
itg_data.domain, itg_data.subdomain_id, {},
None)
sorted_integrals[rule] = integral_new
# TODO: See if coefficient_numbering can be removed
# Build coefficient numbering for UFC interface here, to avoid
# renumbering in UFL and application of replace mapping
coefficient_numbering = {}
for i, f in enumerate(form_data.reduced_coefficients):
coefficient_numbering[f] = i
# Add coefficient numbering to IR
ir["coefficient_numbering"] = coefficient_numbering
index_to_coeff = sorted([(v, k)
for k, v in coefficient_numbering.items()])
offsets = {}
width = 2 if integral_type in ("interior_facet") else 1
_offset = 0
for k, el in zip(index_to_coeff, form_data.coefficient_elements):
offsets[k[1]] = _offset
_offset += width * ir["element_dimensions"][el]
# Copy offsets also into IR
ir["coefficient_offsets"] = offsets
# Build offsets for Constants
original_constant_offsets = {}
_offset = 0
for constant in form_data.original_form.constants():
original_constant_offsets[constant] = _offset
_offset += numpy.product(constant.ufl_shape, dtype=numpy.int)
ir["original_constant_offsets"] = original_constant_offsets
ir["precision"] = itg_data.metadata["precision"]
# Create map from number of quadrature points -> integrand
integrands = {
rule: integral.integrand()
for rule, integral in sorted_integrals.items()
}
# Build more specific intermediate representation
integral_ir = compute_integral_ir(itg_data.domain.ufl_cell(),
itg_data.integral_type,
ir["entitytype"], integrands,
ir["tensor_shape"], parameters,
visualise)
ir.update(integral_ir)
# Fetch name
ir["name"] = integral_names[(form_index, itg_data_index)]
irs.append(ir_integral(**ir))
return irs
def get_ffcx_table_values(points, cell, integral_type, ufl_element, avg,
entitytype, derivative_counts, flat_component):
"""Extract values from ffcx element table.
Returns a 3D numpy array with axes
(entity number, quadrature point number, dof number)
"""
deriv_order = sum(derivative_counts)
if integral_type in ufl.custom_integral_types:
# Use quadrature points on cell for analysis in custom integral types
integral_type = "cell"
assert not avg
if integral_type == "expression":
# FFCX tables for expression are generated as interior cell points
integral_type = "cell"
if avg in ("cell", "facet"):
# Redefine points to compute average tables
# Make sure this is not called with points, that doesn't make sense
# assert points is None
# Not expecting derivatives of averages
assert not any(derivative_counts)
assert deriv_order == 0
# Doesn't matter if it's exterior or interior facet integral,
# just need a valid integral type to create quadrature rule
if avg == "cell":
integral_type = "cell"
elif avg == "facet":
integral_type = "exterior_facet"
# Make quadrature rule and get points and weights
points, weights = create_quadrature_points_and_weights(
integral_type, cell, ufl_element.degree(), "default")
# Tabulate table of basis functions and derivatives in points for each entity
tdim = cell.topological_dimension()
entity_dim = integral_type_to_entity_dim(integral_type, tdim)
num_entities = ufl.cell.num_cell_entities[cell.cellname()][entity_dim]
fiat_element = create_element(ufl_element)
# Extract arrays for the right scalar component
component_tables = []
sh = ufl_element.value_shape()
if sh == ():
# Scalar valued element
for entity in range(num_entities):
entity_points = map_integral_points(points, integral_type, cell,
entity)
tbl = fiat_element.tabulate(deriv_order,
entity_points)[derivative_counts]
component_tables.append(tbl)
elif len(sh) > 0 and ufl_element.num_sub_elements() == 0:
# 2-tensor-valued elements, not a tensor product
# mapping flat_component back to tensor component
(_, f2t) = ufl.permutation.build_component_numbering(
sh, ufl_element.symmetry())
t_comp = f2t[flat_component]
for entity in range(num_entities):
entity_points = map_integral_points(points, integral_type, cell,
entity)
tbl = fiat_element.tabulate(deriv_order,
entity_points)[derivative_counts]
if len(sh) == 1:
component_tables.append(tbl[:, t_comp[0], :])
elif len(sh) == 2:
component_tables.append(tbl[:, t_comp[0], t_comp[1], :])
else:
raise RuntimeError(
"Cannot tabulate tensor valued element with rank > 2")
else:
# Vector-valued or mixed element
sub_dims = [0] + list(e.space_dimension()
for e in fiat_element.elements())
sub_cmps = [0] + list(
numpy.prod(e.value_shape(), dtype=int)
for e in fiat_element.elements())
irange = numpy.cumsum(sub_dims)
crange = numpy.cumsum(sub_cmps)
# Find index of sub element which corresponds to the current flat component
component_element_index = numpy.where(
crange <= flat_component)[0].shape[0] - 1
ir = irange[component_element_index:component_element_index + 2]
cr = crange[component_element_index:component_element_index + 2]
component_element = fiat_element.elements()[component_element_index]
# Get the block size to switch XXYYZZ ordering to XYZXYZ
if isinstance(ufl_element, ufl.VectorElement) or isinstance(
ufl_element, ufl.TensorElement):
block_size = fiat_element.num_sub_elements()
ir = [ir[0] * block_size // irange[-1], irange[-1], block_size]
def slice_size(r):
if len(r) == 1:
return r[0]
if len(r) == 2:
return r[1] - r[0]
if len(r) == 3:
return 1 + (r[1] - r[0] - 1) // r[2]
# Follows from FIAT's MixedElement tabulation
# Tabulating MixedElement in FIAT would result in tabulated subelements
# padded with zeros
for entity in range(num_entities):
entity_points = map_integral_points(points, integral_type, cell,
entity)
# Tabulate subelement, this is dense nonzero table, [a, b, c]
tbl = component_element.tabulate(deriv_order,
entity_points)[derivative_counts]
# Prepare a padded table with zeros
padded_shape = (fiat_element.space_dimension(
), ) + fiat_element.value_shape() + (len(entity_points), )
padded_tbl = numpy.zeros(padded_shape, dtype=tbl.dtype)
tab = tbl.reshape(slice_size(ir), slice_size(cr), -1)
padded_tbl[slice(*ir), slice(*cr)] = tab
component_tables.append(padded_tbl[:, flat_component, :])
if avg in ("cell", "facet"):
# Compute numeric integral of the each component table
wsum = sum(weights)
for entity, tbl in enumerate(component_tables):
num_dofs = tbl.shape[0]
tbl = numpy.dot(tbl, weights) / wsum
tbl = numpy.reshape(tbl, (num_dofs, 1))
component_tables[entity] = tbl
# Loop over entities and fill table blockwise (each block = points x dofs)
# Reorder axes as (points, dofs) instead of (dofs, points)
assert len(component_tables) == num_entities
num_dofs, num_points = component_tables[0].shape
shape = (num_entities, num_points, num_dofs)
res = numpy.zeros(shape)
for entity in range(num_entities):
res[entity, :, :] = numpy.transpose(component_tables[entity])
return res
def get_ffcx_table_values(points, cell, integral_type, ufl_element, avg, entitytype,
derivative_counts, flat_component):
"""Extract values from FFCx element table.
Returns a 3D numpy array with axes
(entity number, quadrature point number, dof number)
"""
deriv_order = sum(derivative_counts)
if integral_type in ufl.custom_integral_types:
# Use quadrature points on cell for analysis in custom integral types
integral_type = "cell"
assert not avg
if integral_type == "expression":
# FFCx tables for expression are generated as interior cell points
integral_type = "cell"
if avg in ("cell", "facet"):
# Redefine points to compute average tables
# Make sure this is not called with points, that doesn't make sense
# assert points is None
# Not expecting derivatives of averages
assert not any(derivative_counts)
assert deriv_order == 0
# Doesn't matter if it's exterior or interior facet integral,
# just need a valid integral type to create quadrature rule
if avg == "cell":
integral_type = "cell"
elif avg == "facet":
integral_type = "exterior_facet"
# Make quadrature rule and get points and weights
points, weights = create_quadrature_points_and_weights(integral_type, cell,
ufl_element.degree(), "default")
# Tabulate table of basis functions and derivatives in points for each entity
tdim = cell.topological_dimension()
entity_dim = integral_type_to_entity_dim(integral_type, tdim)
num_entities = ufl.cell.num_cell_entities[cell.cellname()][entity_dim]
numpy.set_printoptions(suppress=True, precision=2)
basix_element = create_element(ufl_element)
# Extract arrays for the right scalar component
component_tables = []
sh = tuple(basix_element.value_shape)
assert len(sh) > 0
component_element, offset, stride = basix_element.get_component_element(flat_component)
for entity in range(num_entities):
entity_points = map_integral_points(points, integral_type, cell, entity)
tbl = component_element.tabulate(deriv_order, entity_points)
tbl = tbl[basix_index(*derivative_counts)]
component_tables.append(tbl)
if avg in ("cell", "facet"):
# Compute numeric integral of the each component table
wsum = sum(weights)
for entity, tbl in enumerate(component_tables):
num_dofs = tbl.shape[1]
tbl = numpy.dot(tbl, weights) / wsum
tbl = numpy.reshape(tbl, (1, num_dofs))
component_tables[entity] = tbl
# Loop over entities and fill table blockwise (each block = points x dofs)
# Reorder axes as (points, dofs) instead of (dofs, points)
assert len(component_tables) == num_entities
num_points, num_dofs = component_tables[0].shape
shape = (1, num_entities, num_points, num_dofs)
res = numpy.zeros(shape)
for entity in range(num_entities):
res[:, entity, :, :] = component_tables[entity]
return {'array': res, 'offset': offset, 'stride': stride}