def dual_basis(self):
# Return Q with x.indices already a free index for the
# consumer to use
# expensive numerical extraction is done once per element
# instance, but the point set must be created every time we
# build the dual.
Q, pts = self._dual_basis
x = PointSet(pts)
assert len(x.indices) == 1
assert Q.shape[1] == x.indices[0].extent
i, *js = gem.indices(len(Q.shape) - 1)
Q = gem.ComponentTensor(gem.Indexed(Q, (i, *x.indices, *js)), (i, *js))
return Q, x
def factor_point_set(product_cell, product_dim, point_set):
"""Factors a point set for the product element into a point sets for
each subelement.
:arg product_cell: a TensorProductCell
:arg product_dim: entity dimension for the product cell
:arg point_set: point set for the product element
"""
assert len(product_cell.cells) == len(product_dim)
point_dims = [
cell.construct_subelement(dim).get_spatial_dimension()
for cell, dim in zip(product_cell.cells, product_dim)
]
if isinstance(point_set, TensorPointSet):
# Just give the factors asserting matching dimensions.
assert len(point_set.factors) == len(point_dims)
assert all(ps.dimension == dim
for ps, dim in zip(point_set.factors, point_dims))
return point_set.factors
# Split the point coordinates along the point dimensions
# required by the subelements.
assert point_set.dimension == sum(point_dims)
slices = TensorProductCell._split_slices(point_dims)
if isinstance(point_set, PointSingleton):
return [PointSingleton(point_set.point[s]) for s in slices]
elif isinstance(point_set, PointSet):
# Use the same point index for the new point sets.
result = []
for s in slices:
ps = PointSet(point_set.points[:, s])
ps.indices = point_set.indices
result.append(ps)
return result
raise NotImplementedError("How to tabulate TensorProductElement on %s?" %
(type(point_set).__name__, ))
def translate_cell_vertices(terminal, mt, ctx):
coords = SpatialCoordinate(terminal.ufl_domain())
ufl_expr = construct_modified_terminal(mt, coords)
ps = PointSet(numpy.array(ctx.fiat_cell.get_vertices()))
config = {name: getattr(ctx, name)
for name in ["ufl_cell", "precision", "index_cache"]}
config.update(interface=ctx, point_set=ps)
context = PointSetContext(**config)
expr = context.translator(ufl_expr)
# Wrap up point (vertex) index
c = gem.Index()
return gem.ComponentTensor(gem.Indexed(expr, (c,)), ps.indices + (c,))
def factor_point_set(product_cell, product_dim, point_set):
"""Factors a point set for the product element into a point sets for
each subelement.
:arg product_cell: a TensorProductCell
:arg product_dim: entity dimension for the product cell
:arg point_set: point set for the product element
"""
assert len(product_cell.cells) == len(product_dim)
point_dims = [cell.construct_subelement(dim).get_spatial_dimension()
for cell, dim in zip(product_cell.cells, product_dim)]
if isinstance(point_set, TensorPointSet):
# Just give the factors asserting matching dimensions.
assert len(point_set.factors) == len(point_dims)
assert all(ps.dimension == dim
for ps, dim in zip(point_set.factors, point_dims))
return point_set.factors
# Split the point coordinates along the point dimensions
# required by the subelements.
assert point_set.dimension == sum(point_dims)
slices = TensorProductCell._split_slices(point_dims)
if isinstance(point_set, PointSingleton):
return [PointSingleton(point_set.point[s]) for s in slices]
elif isinstance(point_set, PointSet):
# Use the same point index for the new point sets.
result = []
for s in slices:
ps = PointSet(point_set.points[:, s])
ps.indices = point_set.indices
result.append(ps)
return result
raise NotImplementedError("How to tabulate TensorProductElement on %s?" % (type(point_set).__name__,))
def make_quadrature(ref_el, degree, scheme="default"):
"""
Generate quadrature rule for given reference element
that will integrate an polynomial of order 'degree' exactly.
For low-degree (<=6) polynomials on triangles and tetrahedra, this
uses hard-coded rules, otherwise it falls back to a collapsed
Gauss scheme on simplices. On tensor-product cells, it is a
tensor-product quadrature rule of the subcells.
:arg ref_el: The FIAT cell to create the quadrature for.
:arg degree: The degree of polynomial that the rule should
integrate exactly.
"""
if ref_el.get_shape() == TENSORPRODUCT:
try:
degree = tuple(degree)
except TypeError:
degree = (degree, ) * len(ref_el.cells)
assert len(ref_el.cells) == len(degree)
quad_rules = [
make_quadrature(c, d, scheme)
for c, d in zip(ref_el.cells, degree)
]
return TensorProductQuadratureRule(quad_rules)
if ref_el.get_shape() == QUADRILATERAL:
return make_quadrature(ref_el.product, degree, scheme)
if degree < 0:
raise ValueError("Need positive degree, not %d" % degree)
if ref_el.get_shape() == LINE:
# FIAT uses Gauss-Legendre line quadature, however, since we
# symbolically label it as such, we wish not to risk attaching
# the wrong label in case FIAT changes. So we explicitly ask
# for Gauss-Legendre line quadature.
num_points = (degree + 1 + 1) // 2 # exact integration
fiat_rule = GaussLegendreQuadratureLineRule(ref_el, num_points)
point_set = GaussLegendrePointSet(fiat_rule.get_points())
return QuadratureRule(point_set, fiat_rule.get_weights())
fiat_rule = fiat_scheme(ref_el, degree, scheme)
return QuadratureRule(PointSet(fiat_rule.get_points()),
fiat_rule.get_weights())
def compile_expression_at_points(expression, points, coordinates, parameters=None):
"""Compiles a UFL expression to be evaluated at compile-time known
reference points. Useful for interpolating UFL expressions onto
function spaces with only point evaluation nodes.
:arg expression: UFL expression
:arg points: reference coordinates of the evaluation points
:arg coordinates: the coordinate function
:arg parameters: parameters object
"""
import coffee.base as ast
if parameters is None:
parameters = default_parameters()
else:
_ = default_parameters()
_.update(parameters)
parameters = _
# No arguments, please!
if extract_arguments(expression):
return ValueError("Cannot interpolate UFL expression with Arguments!")
# Apply UFL preprocessing
expression = ufl_utils.preprocess_expression(expression)
# Initialise kernel builder
builder = firedrake_interface.ExpressionKernelBuilder()
# Replace coordinates (if any)
domain = expression.ufl_domain()
if domain:
assert coordinates.ufl_domain() == domain
builder.domain_coordinate[domain] = coordinates
# Collect required coefficients
coefficients = extract_coefficients(expression)
if has_type(expression, GeometricQuantity):
coefficients = [coordinates] + coefficients
builder.set_coefficients(coefficients)
# Split mixed coefficients
expression = ufl_utils.split_coefficients(expression, builder.coefficient_split)
# Translate to GEM
point_set = PointSet(points)
config = dict(interface=builder,
ufl_cell=coordinates.ufl_domain().ufl_cell(),
precision=parameters["precision"],
point_set=point_set)
ir, = fem.compile_ufl(expression, point_sum=False, **config)
# Deal with non-scalar expressions
value_shape = ir.shape
tensor_indices = tuple(gem.Index() for s in value_shape)
if value_shape:
ir = gem.Indexed(ir, tensor_indices)
# Build kernel body
return_shape = (len(points),) + value_shape
return_indices = point_set.indices + tensor_indices
return_var = gem.Variable('A', return_shape)
return_arg = ast.Decl(SCALAR_TYPE, ast.Symbol('A', rank=return_shape))
return_expr = gem.Indexed(return_var, return_indices)
ir, = impero_utils.preprocess_gem([ir])
impero_c = impero_utils.compile_gem([(return_expr, ir)], return_indices)
point_index, = point_set.indices
body = generate_coffee(impero_c, {point_index: 'p'}, parameters["precision"])
# Handle cell orientations
if builder.needs_cell_orientations([ir]):
builder.require_cell_orientations()
# Build kernel tuple
return builder.construct_kernel(return_arg, body)
def point_set(self):
return PointSet(self._points)
def physical_vertices(self):
vs = PointSet(self.interface.fiat_cell.vertices)
return self.physical_points(vs)
def compile_expression_at_points(expression,
points,
coordinates,
interface=None,
parameters=None,
coffee=True):
"""Compiles a UFL expression to be evaluated at compile-time known
reference points. Useful for interpolating UFL expressions onto
function spaces with only point evaluation nodes.
:arg expression: UFL expression
:arg points: reference coordinates of the evaluation points
:arg coordinates: the coordinate function
:arg interface: backend module for the kernel interface
:arg parameters: parameters object
:arg coffee: compile coffee kernel instead of loopy kernel
"""
import coffee.base as ast
import loopy as lp
if parameters is None:
parameters = default_parameters()
else:
_ = default_parameters()
_.update(parameters)
parameters = _
# Determine whether in complex mode
complex_mode = is_complex(parameters["scalar_type"])
# Apply UFL preprocessing
expression = ufl_utils.preprocess_expression(expression,
complex_mode=complex_mode)
# Initialise kernel builder
if interface is None:
if coffee:
import tsfc.kernel_interface.firedrake as firedrake_interface_coffee
interface = firedrake_interface_coffee.ExpressionKernelBuilder
else:
# Delayed import, loopy is a runtime dependency
import tsfc.kernel_interface.firedrake_loopy as firedrake_interface_loopy
interface = firedrake_interface_loopy.ExpressionKernelBuilder
builder = interface(parameters["scalar_type"])
arguments = extract_arguments(expression)
argument_multiindices = tuple(
builder.create_element(arg.ufl_element()).get_indices()
for arg in arguments)
# Replace coordinates (if any)
domain = expression.ufl_domain()
if domain:
assert coordinates.ufl_domain() == domain
builder.domain_coordinate[domain] = coordinates
builder.set_cell_sizes(domain)
# Collect required coefficients
coefficients = extract_coefficients(expression)
if has_type(expression, GeometricQuantity) or any(
fem.needs_coordinate_mapping(c.ufl_element())
for c in coefficients):
coefficients = [coordinates] + coefficients
builder.set_coefficients(coefficients)
# Split mixed coefficients
expression = ufl_utils.split_coefficients(expression,
builder.coefficient_split)
# Translate to GEM
point_set = PointSet(points)
config = dict(interface=builder,
ufl_cell=coordinates.ufl_domain().ufl_cell(),
precision=parameters["precision"],
point_set=point_set,
argument_multiindices=argument_multiindices)
ir, = fem.compile_ufl(expression, point_sum=False, **config)
# Deal with non-scalar expressions
value_shape = ir.shape
tensor_indices = tuple(gem.Index() for s in value_shape)
if value_shape:
ir = gem.Indexed(ir, tensor_indices)
# Build kernel body
return_indices = point_set.indices + tensor_indices + tuple(
chain(*argument_multiindices))
return_shape = tuple(i.extent for i in return_indices)
return_var = gem.Variable('A', return_shape)
if coffee:
return_arg = ast.Decl(parameters["scalar_type"],
ast.Symbol('A', rank=return_shape))
else:
return_arg = lp.GlobalArg("A",
dtype=parameters["scalar_type"],
shape=return_shape)
return_expr = gem.Indexed(return_var, return_indices)
ir, = impero_utils.preprocess_gem([ir])
impero_c = impero_utils.compile_gem([(return_expr, ir)], return_indices)
point_index, = point_set.indices
# Handle kernel interface requirements
builder.register_requirements([ir])
# Build kernel tuple
return builder.construct_kernel(return_arg, impero_c,
parameters["precision"],
{point_index: 'p'})
def ps(cell):
if cell == "tet":
return PointSet([[1 / 3, 1 / 4, 1 / 5]])
elif cell == "quad":
return PointSet([[1 / 3, 1 / 4]])
def compile_expression_dual_evaluation(expression,
to_element,
*,
domain=None,
interface=None,
parameters=None,
coffee=False):
"""Compile a UFL expression to be evaluated against a compile-time known reference element's dual basis.
Useful for interpolating UFL expressions into e.g. N1curl spaces.
:arg expression: UFL expression
:arg to_element: A FInAT element for the target space
:arg domain: optional UFL domain the expression is defined on (required when expression contains no domain).
:arg interface: backend module for the kernel interface
:arg parameters: parameters object
:arg coffee: compile coffee kernel instead of loopy kernel
"""
import coffee.base as ast
import loopy as lp
# Just convert FInAT element to FIAT for now.
# Dual evaluation in FInAT will bring a thorough revision.
to_element = to_element.fiat_equivalent
if any(len(dual.deriv_dict) != 0 for dual in to_element.dual_basis()):
raise NotImplementedError(
"Can only interpolate onto dual basis functionals without derivative evaluation, sorry!"
)
if parameters is None:
parameters = default_parameters()
else:
_ = default_parameters()
_.update(parameters)
parameters = _
# Determine whether in complex mode
complex_mode = is_complex(parameters["scalar_type"])
# Find out which mapping to apply
try:
mapping, = set(to_element.mapping())
except ValueError:
raise NotImplementedError(
"Don't know how to interpolate onto zany spaces, sorry")
expression = apply_mapping(expression, mapping, domain)
# Apply UFL preprocessing
expression = ufl_utils.preprocess_expression(expression,
complex_mode=complex_mode)
# Initialise kernel builder
if interface is None:
if coffee:
import tsfc.kernel_interface.firedrake as firedrake_interface_coffee
interface = firedrake_interface_coffee.ExpressionKernelBuilder
else:
# Delayed import, loopy is a runtime dependency
import tsfc.kernel_interface.firedrake_loopy as firedrake_interface_loopy
interface = firedrake_interface_loopy.ExpressionKernelBuilder
builder = interface(parameters["scalar_type"])
arguments = extract_arguments(expression)
argument_multiindices = tuple(
builder.create_element(arg.ufl_element()).get_indices()
for arg in arguments)
# Replace coordinates (if any) unless otherwise specified by kwarg
if domain is None:
domain = expression.ufl_domain()
assert domain is not None
# Collect required coefficients
first_coefficient_fake_coords = False
coefficients = extract_coefficients(expression)
if has_type(expression, GeometricQuantity) or any(
fem.needs_coordinate_mapping(c.ufl_element())
for c in coefficients):
# Create a fake coordinate coefficient for a domain.
coords_coefficient = ufl.Coefficient(
ufl.FunctionSpace(domain, domain.ufl_coordinate_element()))
builder.domain_coordinate[domain] = coords_coefficient
builder.set_cell_sizes(domain)
coefficients = [coords_coefficient] + coefficients
first_coefficient_fake_coords = True
builder.set_coefficients(coefficients)
# Split mixed coefficients
expression = ufl_utils.split_coefficients(expression,
builder.coefficient_split)
# Translate to GEM
kernel_cfg = dict(
interface=builder,
ufl_cell=domain.ufl_cell(),
# FIXME: change if we ever implement
# interpolation on facets.
integral_type="cell",
argument_multiindices=argument_multiindices,
index_cache={},
scalar_type=parameters["scalar_type"])
if all(
isinstance(dual, PointEvaluation)
for dual in to_element.dual_basis()):
# This is an optimisation for point-evaluation nodes which
# should go away once FInAT offers the interface properly
qpoints = []
# Everything is just a point evaluation.
for dual in to_element.dual_basis():
ptdict = dual.get_point_dict()
qpoint, = ptdict.keys()
(qweight, component), = ptdict[qpoint]
assert allclose(qweight, 1.0)
assert component == ()
qpoints.append(qpoint)
point_set = PointSet(qpoints)
config = kernel_cfg.copy()
config.update(point_set=point_set)
# Allow interpolation onto QuadratureElements to refer to the quadrature
# rule they represent
if isinstance(to_element, FIAT.QuadratureElement):
assert allclose(asarray(qpoints), asarray(to_element._points))
quad_rule = QuadratureRule(point_set, to_element._weights)
config["quadrature_rule"] = quad_rule
expr, = fem.compile_ufl(expression, **config, point_sum=False)
# In some cases point_set.indices may be dropped from expr, but nothing
# new should now appear
assert set(expr.free_indices) <= set(
chain(point_set.indices, *argument_multiindices))
shape_indices = tuple(gem.Index() for _ in expr.shape)
basis_indices = point_set.indices
ir = gem.Indexed(expr, shape_indices)
else:
# This is general code but is more unrolled than necssary.
dual_expressions = [] # one for each functional
broadcast_shape = len(expression.ufl_shape) - len(
to_element.value_shape())
shape_indices = tuple(gem.Index()
for _ in expression.ufl_shape[:broadcast_shape])
expr_cache = {} # Sharing of evaluation of the expression at points
for dual in to_element.dual_basis():
pts = tuple(sorted(dual.get_point_dict().keys()))
try:
expr, point_set = expr_cache[pts]
except KeyError:
point_set = PointSet(pts)
config = kernel_cfg.copy()
config.update(point_set=point_set)
expr, = fem.compile_ufl(expression, **config, point_sum=False)
# In some cases point_set.indices may be dropped from expr, but
# nothing new should now appear
assert set(expr.free_indices) <= set(
chain(point_set.indices, *argument_multiindices))
expr = gem.partial_indexed(expr, shape_indices)
expr_cache[pts] = expr, point_set
weights = collections.defaultdict(list)
for p in pts:
for (w, cmp) in dual.get_point_dict()[p]:
weights[cmp].append(w)
qexprs = gem.Zero()
for cmp in sorted(weights):
qweights = gem.Literal(weights[cmp])
qexpr = gem.Indexed(expr, cmp)
qexpr = gem.index_sum(
gem.Indexed(qweights, point_set.indices) * qexpr,
point_set.indices)
qexprs = gem.Sum(qexprs, qexpr)
assert qexprs.shape == ()
assert set(qexprs.free_indices) == set(
chain(shape_indices, *argument_multiindices))
dual_expressions.append(qexprs)
basis_indices = (gem.Index(), )
ir = gem.Indexed(gem.ListTensor(dual_expressions), basis_indices)
# Build kernel body
return_indices = basis_indices + shape_indices + tuple(
chain(*argument_multiindices))
return_shape = tuple(i.extent for i in return_indices)
return_var = gem.Variable('A', return_shape)
if coffee:
return_arg = ast.Decl(parameters["scalar_type"],
ast.Symbol('A', rank=return_shape))
else:
return_arg = lp.GlobalArg("A",
dtype=parameters["scalar_type"],
shape=return_shape)
return_expr = gem.Indexed(return_var, return_indices)
# TODO: one should apply some GEM optimisations as in assembly,
# but we don't for now.
ir, = impero_utils.preprocess_gem([ir])
impero_c = impero_utils.compile_gem([(return_expr, ir)], return_indices)
index_names = dict(
(idx, "p%d" % i) for (i, idx) in enumerate(basis_indices))
# Handle kernel interface requirements
builder.register_requirements([ir])
# Build kernel tuple
return builder.construct_kernel(return_arg, impero_c, index_names,
first_coefficient_fake_coords)