def basis_evaluation(self, order, ps, entity=None):
'''Return code for evaluating the element at known points on the
reference element.
:param order: return derivatives up to this order.
:param ps: the point set.
:param entity: the cell entity on which to tabulate.
'''
space_dimension = self._element.space_dimension()
value_size = np.prod(self._element.value_shape(), dtype=int)
fiat_result = self._element.tabulate(order, ps.points, entity)
result = {}
for alpha, fiat_table in iteritems(fiat_result):
if isinstance(fiat_table, Exception):
result[alpha] = gem.Failure(
self.index_shape + self.value_shape, fiat_table)
continue
derivative = sum(alpha)
table_roll = fiat_table.reshape(space_dimension, value_size,
len(ps.points)).transpose(1, 2, 0)
exprs = []
for table in table_roll:
if derivative < self.degree:
point_indices = ps.indices
point_shape = tuple(index.extent
for index in point_indices)
exprs.append(
gem.partial_indexed(
gem.Literal(
table.reshape(point_shape + self.index_shape)),
point_indices))
elif derivative == self.degree:
# Make sure numerics satisfies theory
assert np.allclose(table, table.mean(axis=0,
keepdims=True))
exprs.append(gem.Literal(table[0]))
else:
# Make sure numerics satisfies theory
assert np.allclose(table, 0.0)
exprs.append(gem.Zero(self.index_shape))
if self.value_shape:
beta = self.get_indices()
zeta = self.get_value_indices()
result[alpha] = gem.ComponentTensor(
gem.Indexed(
gem.ListTensor(
np.array([
gem.Indexed(expr, beta) for expr in exprs
]).reshape(self.value_shape)), zeta), beta + zeta)
else:
expr, = exprs
result[alpha] = expr
return result
def test_pickle_gem(protocol):
f = gem.VariableIndex(gem.Indexed(gem.Variable('facet', (2, )), (1, )))
q = gem.Index()
r = gem.Index()
_1 = gem.Indexed(gem.Literal(numpy.random.rand(3, 6, 8)), (f, q, r))
_2 = gem.Indexed(
gem.view(gem.Variable('w', (None, None)), slice(8), slice(1)), (r, 0))
expr = gem.ComponentTensor(gem.IndexSum(gem.Product(_1, _2), (r, )), (q, ))
unpickled = pickle.loads(pickle.dumps(expr, protocol))
assert repr(expr) == repr(unpickled)
def dual_basis(self):
Q, x = self.wrappee.dual_basis
beta = self.get_indices()
zeta = self.get_value_indices()
# Index out the basis indices from wrapee's Q, to get
# something of wrappee.value_shape, then promote to new shape
# with the same transform as done for basis evaluation
Q = gem.ListTensor(self.transform(gem.partial_indexed(Q, beta)))
# Finally wrap up Q in shape again (now with some extra
# value_shape indices)
return gem.ComponentTensor(Q[zeta], beta + zeta), x
def basis_evaluation(self, order, ps, entity=None):
# Default entity
if entity is None:
entity = (self.cell.get_dimension(), 0)
entity_dim, entity_id = entity
# Factor entity
assert isinstance(entity_dim, tuple)
assert len(entity_dim) == len(self.factors)
shape = tuple(
len(c.get_topology()[d])
for c, d in zip(self.cell.cells, entity_dim))
entities = list(zip(entity_dim, numpy.unravel_index(entity_id, shape)))
# Factor point set
ps_factors = factor_point_set(self.cell, entity_dim, ps)
# Subelement results
factor_results = [
fe.basis_evaluation(order, ps_, e)
for fe, ps_, e in zip(self.factors, ps_factors, entities)
]
# Spatial dimension
dimension = self.cell.get_spatial_dimension()
# A list of slices that are used to select dimensions
# corresponding to each subelement.
dim_slices = TensorProductCell._split_slices(
[c.get_spatial_dimension() for c in self.cell.cells])
# A list of multiindices, one multiindex per subelement, each
# multiindex describing the shape of basis functions of the
# subelement.
alphas = [fe.get_indices() for fe in self.factors]
result = {}
for derivative in range(order + 1):
for Delta in mis(dimension, derivative):
# Split the multiindex for the subelements
deltas = [Delta[s] for s in dim_slices]
# GEM scalars (can have free indices) for collecting
# the contributions from the subelements.
scalars = []
for fr, delta, alpha in zip(factor_results, deltas, alphas):
# Turn basis shape to free indices, select the
# right derivative entry, and collect the result.
scalars.append(gem.Indexed(fr[delta], alpha))
# Multiply the values from the subelements and wrap up
# non-point indices into shape.
result[Delta] = gem.ComponentTensor(
reduce(gem.Product, scalars), tuple(chain(*alphas)))
return result
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 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 translate_cell_edge_vectors(terminal, mt, ctx):
# WARNING: Assumes straight edges!
coords = CellVertices(terminal.ufl_domain())
ufl_expr = construct_modified_terminal(mt, coords)
cell_vertices = ctx.translator(ufl_expr)
e = gem.Index()
c = gem.Index()
expr = gem.ListTensor([
gem.Sum(
gem.Indexed(cell_vertices, (u, c)),
gem.Product(gem.Literal(-1), gem.Indexed(cell_vertices, (v, c))))
for _, (u, v) in sorted(ctx.fiat_cell.get_topology()[1].items())
])
return gem.ComponentTensor(gem.Indexed(expr, (e, )), (e, c))
def dual_basis(self):
# Outer product the dual bases of the factors
qs, pss = zip(*(factor.dual_basis for factor in self.factors))
ps = TensorPointSet(pss)
# Naming as _merge_evaluations above
alphas = [factor.get_indices() for factor in self.factors]
zetas = [factor.get_value_indices() for factor in self.factors]
# Index the factors by so that we can reshape into index-shape
# followed by value-shape
qis = [q[alpha + zeta] for q, alpha, zeta in zip(qs, alphas, zetas)]
Q = gem.ComponentTensor(
reduce(gem.Product, qis),
tuple(chain(*(alphas + zetas)))
)
return Q, ps
def basis_evaluation(self, order, ps, entity=None):
'''Return code for evaluating the element at known points on the
reference element.
:param order: return derivatives up to this order.
:param ps: the point set.
:param entity: the cell entity on which to tabulate.
'''
result = super(GaussLegendre, self).basis_evaluation(order, ps, entity)
cell_dimension = self.cell.get_dimension()
if entity is None or entity == (cell_dimension, 0): # on cell interior
space_dim = self.space_dimension()
if isinstance(ps, GaussLegendrePointSet) and len(ps.points) == space_dim:
# Bingo: evaluation points match node locations!
spatial_dim = self.cell.get_spatial_dimension()
q, = ps.indices
r, = self.get_indices()
result[(0,) * spatial_dim] = gem.ComponentTensor(gem.Delta(q, r), (r,))
return result
def entity_support_dofs(elem, entity_dim):
"""Return the map of entity id to the degrees of freedom for which
the corresponding basis functions take non-zero values.
:arg elem: FInAT finite element
:arg entity_dim: Dimension of the cell subentity.
"""
if not hasattr(elem, "_entity_support_dofs"):
elem._entity_support_dofs = {}
cache = elem._entity_support_dofs
try:
return cache[entity_dim]
except KeyError:
pass
beta = elem.get_indices()
zeta = elem.get_value_indices()
entity_cell = elem.cell.construct_subelement(entity_dim)
quad = make_quadrature(entity_cell,
(2 * numpy.array(elem.degree)).tolist())
eps = 1.e-8 # Is this a safe value?
result = {}
for f in elem.entity_dofs()[entity_dim].keys():
# Tabulate basis functions on the facet
vals, = itervalues(
elem.basis_evaluation(0, quad.point_set, entity=(entity_dim, f)))
# Integrate the square of the basis functions on the facet.
ints = gem.IndexSum(
gem.Product(
gem.IndexSum(
gem.Product(gem.Indexed(vals, beta + zeta),
gem.Indexed(vals, beta + zeta)), zeta),
quad.weight_expression), quad.point_set.indices)
ints = aggressive_unroll(gem.ComponentTensor(ints,
beta)).array.flatten()
result[f] = [dof for dof, i in enumerate(ints) if i > eps]
cache[entity_dim] = result
return result
def point_evaluation_generic(fiat_element, order, refcoords, entity):
# Coordinates on the reference entity (SymPy)
esd, = refcoords.shape
Xi = sp.symbols('X Y Z')[:esd]
space_dimension = fiat_element.space_dimension()
value_size = np.prod(fiat_element.value_shape(), dtype=int)
fiat_result = fiat_element.tabulate(order, [Xi], entity)
result = {}
for alpha, fiat_table in fiat_result.items():
if isinstance(fiat_table, Exception):
result[alpha] = gem.Failure(
(space_dimension, ) + fiat_element.value_shape(), fiat_table)
continue
# Convert SymPy expression to GEM
mapper = gem.node.Memoizer(sympy2gem)
mapper.bindings = {
s: gem.Indexed(refcoords, (i, ))
for i, s in enumerate(Xi)
}
gem_table = np.vectorize(mapper)(fiat_table)
table_roll = gem_table.reshape(space_dimension, value_size).transpose()
exprs = []
for table in table_roll:
exprs.append(gem.ListTensor(table.reshape(space_dimension)))
if fiat_element.value_shape():
beta = (gem.Index(extent=space_dimension), )
zeta = tuple(
gem.Index(extent=d) for d in fiat_element.value_shape())
result[alpha] = gem.ComponentTensor(
gem.Indexed(
gem.ListTensor(
np.array([gem.Indexed(expr, beta) for expr in exprs
]).reshape(fiat_element.value_shape())),
zeta), beta + zeta)
else:
expr, = exprs
result[alpha] = expr
return result
def fiat_to_ufl(fiat_dict, order):
# All derivative multiindices must be of the same dimension.
dimension, = set(len(alpha) for alpha in fiat_dict.keys())
# All derivative tables must have the same shape.
shape, = set(table.shape for table in fiat_dict.values())
sigma = tuple(gem.Index(extent=extent) for extent in shape)
# Convert from FIAT to UFL format
eye = numpy.eye(dimension, dtype=int)
tensor = numpy.empty((dimension, ) * order, dtype=object)
for multiindex in numpy.ndindex(tensor.shape):
alpha = tuple(eye[multiindex, :].sum(axis=0))
tensor[multiindex] = gem.Indexed(fiat_dict[alpha], sigma)
delta = tuple(gem.Index(extent=dimension) for _ in range(order))
if order > 0:
tensor = gem.Indexed(gem.ListTensor(tensor), delta)
else:
tensor = tensor[()]
return gem.ComponentTensor(tensor, sigma + delta)
def translate_argument(terminal, mt, ctx):
argument_multiindex = ctx.argument_multiindices[terminal.number()]
sigma = tuple(gem.Index(extent=d) for d in mt.expr.ufl_shape)
element = create_element(terminal.ufl_element())
def callback(entity_id):
finat_dict = ctx.basis_evaluation(element, mt.local_derivatives, entity_id)
# Filter out irrelevant derivatives
filtered_dict = {alpha: table
for alpha, table in iteritems(finat_dict)
if sum(alpha) == mt.local_derivatives}
# Change from FIAT to UFL arrangement
square = fiat_to_ufl(filtered_dict, mt.local_derivatives)
# A numerical hack that FFC used to apply on FIAT tables still
# lives on after ditching FFC and switching to FInAT.
return ffc_rounding(square, ctx.epsilon)
table = ctx.entity_selector(callback, mt.restriction)
return gem.ComponentTensor(gem.Indexed(table, argument_multiindex + sigma), sigma)
def _merge_evaluations(self, factor_results):
# Spatial dimension
dimension = self.cell.get_spatial_dimension()
# Derivative order
order = max(map(sum, chain(*factor_results)))
# A list of slices that are used to select dimensions
# corresponding to each subelement.
dim_slices = TensorProductCell._split_slices([c.get_spatial_dimension()
for c in self.cell.cells])
# A list of multiindices, one multiindex per subelement, each
# multiindex describing the shape of basis functions of the
# subelement.
alphas = [fe.get_indices() for fe in self.factors]
# A list of multiindices, one multiindex per subelement, each
# multiindex describing the value shape of the subelement.
zetas = [fe.get_value_indices() for fe in self.factors]
result = {}
for derivative in range(order + 1):
for Delta in mis(dimension, derivative):
# Split the multiindex for the subelements
deltas = [Delta[s] for s in dim_slices]
# GEM scalars (can have free indices) for collecting
# the contributions from the subelements.
scalars = []
for fr, delta, alpha, zeta in zip(factor_results, deltas, alphas, zetas):
# Turn basis shape to free indices, select the
# right derivative entry, and collect the result.
scalars.append(gem.Indexed(fr[delta], alpha + zeta))
# Multiply the values from the subelements and wrap up
# non-point indices into shape.
result[Delta] = gem.ComponentTensor(
reduce(gem.Product, scalars),
tuple(chain(*(alphas + zetas)))
)
return result
def dual_basis(self):
base = self.base_element
Q, points = base.dual_basis
# Suppose the tensor element has shape (2, 4)
# These identity matrices may have difference sizes depending the shapes
# tQ = Q ⊗ 𝟙₂ ⊗ 𝟙₄
scalar_i = self.base_element.get_indices()
scalar_vi = self.base_element.get_value_indices()
tensor_i = tuple(gem.Index(extent=d) for d in self._shape)
tensor_vi = tuple(gem.Index(extent=d) for d in self._shape)
# Couple new basis function and value indices
deltas = reduce(gem.Product, (gem.Delta(j, k)
for j, k in zip(tensor_i, tensor_vi)))
if self._transpose:
index_ordering = tensor_i + scalar_i + tensor_vi + scalar_vi
else:
index_ordering = scalar_i + tensor_i + tensor_vi + scalar_vi
Qi = Q[scalar_i + scalar_vi]
tQ = gem.ComponentTensor(Qi*deltas, index_ordering)
return tQ, points
def _tensorise(self, scalar_evaluation):
# Old basis function and value indices
scalar_i = self._base_element.get_indices()
scalar_vi = self._base_element.get_value_indices()
# New basis function and value indices
tensor_i = tuple(gem.Index(extent=d) for d in self._shape)
tensor_vi = tuple(gem.Index(extent=d) for d in self._shape)
# Couple new basis function and value indices
deltas = reduce(gem.Product,
(gem.Delta(j, k) for j, k in zip(tensor_i, tensor_vi)))
if self._transpose:
index_ordering = tensor_i + scalar_i + tensor_vi + scalar_vi
else:
index_ordering = scalar_i + tensor_i + tensor_vi + scalar_vi
result = {}
for alpha, expr in scalar_evaluation.items():
result[alpha] = gem.ComponentTensor(
gem.Product(deltas, gem.Indexed(expr, scalar_i + scalar_vi)),
index_ordering)
return result
def basis_evaluation(self,
order,
ps,
entity=None,
coordinate_mapping=None):
'''Return code for evaluating the element at known points on the
reference element.
:param order: return derivatives up to this order.
:param ps: the point set.
:param entity: the cell entity on which to tabulate.
'''
space_dimension = self._element.space_dimension()
value_size = np.prod(self._element.value_shape(), dtype=int)
fiat_result = self._element.tabulate(order, ps.points, entity)
result = {}
# In almost all cases, we have
# self.space_dimension() == self._element.space_dimension()
# But for Bell, FIAT reports 21 basis functions,
# but FInAT only 18 (because there are actually 18
# basis functions, and the additional 3 are for
# dealing with transformations between physical
# and reference space).
index_shape = (self._element.space_dimension(), )
for alpha, fiat_table in fiat_result.items():
if isinstance(fiat_table, Exception):
result[alpha] = gem.Failure(
self.index_shape + self.value_shape, fiat_table)
continue
derivative = sum(alpha)
table_roll = fiat_table.reshape(space_dimension, value_size,
len(ps.points)).transpose(1, 2, 0)
exprs = []
for table in table_roll:
if derivative < self.degree:
point_indices = ps.indices
point_shape = tuple(index.extent
for index in point_indices)
exprs.append(
gem.partial_indexed(
gem.Literal(
table.reshape(point_shape + index_shape)),
point_indices))
elif derivative == self.degree:
# Make sure numerics satisfies theory
exprs.append(gem.Literal(table[0]))
else:
# Make sure numerics satisfies theory
assert np.allclose(table, 0.0)
exprs.append(gem.Zero(self.index_shape))
if self.value_shape:
# As above, this extent may be different from that
# advertised by the finat element.
beta = tuple(gem.Index(extent=i) for i in index_shape)
assert len(beta) == len(self.get_indices())
zeta = self.get_value_indices()
result[alpha] = gem.ComponentTensor(
gem.Indexed(
gem.ListTensor(
np.array([
gem.Indexed(expr, beta) for expr in exprs
]).reshape(self.value_shape)), zeta), beta + zeta)
else:
expr, = exprs
result[alpha] = expr
return result
def sum(self, o, *ops):
shape, = set(o.shape for o in ops)
indices = gem.indices(len(shape))
return gem.ComponentTensor(
gem.Sum(*[gem.Indexed(op, indices) for op in ops]), indices)
def promote(table):
v = gem.partial_indexed(table, beta)
u = gem.ListTensor(self.transform(v))
return gem.ComponentTensor(gem.Indexed(u, zeta), beta + zeta)
def abs(self, o, expr):
indices = gem.indices(len(expr.shape))
return gem.ComponentTensor(
gem.MathFunction('abs', gem.Indexed(expr, indices)), indices)
def index_sum(self, o, summand, indices):
index, = indices
indices = gem.indices(len(summand.shape))
return gem.ComponentTensor(
gem.IndexSum(gem.Indexed(summand, indices), (index, )), indices)
def translate_coefficient(terminal, mt, ctx):
vec = ctx.coefficient(terminal, mt.restriction)
if terminal.ufl_element().family() == 'Real':
assert mt.local_derivatives == 0
return vec
element = ctx.create_element(terminal.ufl_element(),
restriction=mt.restriction)
# Collect FInAT tabulation for all entities
per_derivative = collections.defaultdict(list)
for entity_id in ctx.entity_ids:
finat_dict = ctx.basis_evaluation(element, mt, entity_id)
for alpha, table in finat_dict.items():
# Filter out irrelevant derivatives
if sum(alpha) == mt.local_derivatives:
# A numerical hack that FFC used to apply on FIAT
# tables still lives on after ditching FFC and
# switching to FInAT.
table = ffc_rounding(table, ctx.epsilon)
per_derivative[alpha].append(table)
# Merge entity tabulations for each derivative
if len(ctx.entity_ids) == 1:
def take_singleton(xs):
x, = xs # asserts singleton
return x
per_derivative = {
alpha: take_singleton(tables)
for alpha, tables in per_derivative.items()
}
else:
f = ctx.entity_number(mt.restriction)
per_derivative = {
alpha: gem.select_expression(tables, f)
for alpha, tables in per_derivative.items()
}
# Coefficient evaluation
ctx.index_cache.setdefault(terminal.ufl_element(), element.get_indices())
beta = ctx.index_cache[terminal.ufl_element()]
zeta = element.get_value_indices()
vec_beta, = gem.optimise.remove_componenttensors([gem.Indexed(vec, beta)])
value_dict = {}
for alpha, table in per_derivative.items():
table_qi = gem.Indexed(table, beta + zeta)
summands = []
for var, expr in unconcatenate([(vec_beta, table_qi)],
ctx.index_cache):
indices = tuple(i for i in var.index_ordering()
if i not in ctx.unsummed_coefficient_indices)
value = gem.IndexSum(gem.Product(expr, var), indices)
summands.append(gem.optimise.contraction(value))
optimised_value = gem.optimise.make_sum(summands)
value_dict[alpha] = gem.ComponentTensor(optimised_value, zeta)
# Change from FIAT to UFL arrangement
result = fiat_to_ufl(value_dict, mt.local_derivatives)
assert result.shape == mt.expr.ufl_shape
assert set(result.free_indices) - ctx.unsummed_coefficient_indices <= set(
ctx.point_indices)
# Detect Jacobian of affine cells
if not result.free_indices and all(
numpy.count_nonzero(node.array) <= 2
for node in traversal((result, ))
if isinstance(node, gem.Literal)):
result = gem.optimise.aggressive_unroll(result)
return result
def transpose(expr):
indices = tuple(gem.Index(extent=extent) for extent in expr.shape)
transposed_indices = indices[scalar_rank:] + indices[:scalar_rank]
return gem.ComponentTensor(gem.Indexed(expr, indices),
transposed_indices)
def _dual_basis(self):
# Return the numerical part of the dual basis, this split is
# needed because the dual_basis itself can't produce the same
# point set over and over in case it is used multiple times
# (in for example a tensorproductelement).
fiat_dual_basis = self._element.dual_basis()
seen = dict()
allpts = []
# Find the unique points to evaluate at.
# We might be able to make this a smaller set by treating each
# point one by one, but most of the redundancy comes from
# multiple functionals using the same quadrature rule.
for dual in fiat_dual_basis:
if len(dual.deriv_dict) != 0:
raise NotImplementedError(
"FIAT dual bases with derivative nodes represented via a ``Functional.deriv_dict`` property do not currently have a FInAT dual basis"
)
pts = dual.get_point_dict().keys()
pts = tuple(sorted(pts)) # need this for determinism
if pts not in seen:
# k are indices into Q (see below) for the seen points
kstart = len(allpts)
kend = kstart + len(pts)
seen[pts] = kstart, kend
allpts.extend(pts)
# Build Q.
# Q is a tensor of weights (of total rank R) to contract with a unique
# vector of points to evaluate at, giving a tensor (of total rank R-1)
# where the first indices (rows) correspond to a basis functional
# (node).
# Q is a DOK Sparse matrix in (row, col, higher,..)=>value pairs (to
# become a gem.SparseLiteral when implemented).
# Rows (i) are number of nodes/dual functionals.
# Columns (k) are unique points to evaluate.
# Higher indices (*cmp) are tensor indices of the weights when weights
# are tensor valued.
Q = {}
for i, dual in enumerate(fiat_dual_basis):
point_dict = dual.get_point_dict()
pts = tuple(sorted(point_dict.keys()))
kstart, kend = seen[pts]
for p, k in zip(pts, range(kstart, kend)):
for weight, cmp in point_dict[p]:
Q[(i, k, *cmp)] = weight
if all(
len(set(key)) == 1 and np.isclose(weight, 1) and len(key) == 2
for key, weight in Q.items()):
# Identity matrix Q can be expressed symbolically
extents = tuple(map(max, zip(*Q.keys())))
js = tuple(gem.Index(extent=e + 1) for e in extents)
assert len(js) == 2
Q = gem.ComponentTensor(gem.Delta(*js), js)
else:
# temporary until sparse literals are implemented in GEM which will
# automatically convert a dictionary of keys internally.
# TODO the below is unnecessarily slow and would be sped up
# significantly by building Q in a COO format rather than DOK (i.e.
# storing coords and associated data in (nonzeros, entries) shaped
# numpy arrays) to take advantage of numpy multiindexing
Qshape = tuple(s + 1 for s in map(max, *Q))
Qdense = np.zeros(Qshape, dtype=np.float64)
for idx, value in Q.items():
Qdense[idx] = value
Q = gem.Literal(Qdense)
return Q, np.asarray(allpts)
def dg_injection_kernel(Vf, Vc, ncell):
from firedrake import Tensor, AssembledVector, TestFunction, TrialFunction
from firedrake.slate.slac import compile_expression
macro_builder = MacroKernelBuilder(ScalarType_c, ncell)
f = ufl.Coefficient(Vf)
macro_builder.set_coefficients([f])
macro_builder.set_coordinates(Vf.mesh())
Vfe = create_element(Vf.ufl_element())
macro_quadrature_rule = make_quadrature(
Vfe.cell, estimate_total_polynomial_degree(ufl.inner(f, f)))
index_cache = {}
parameters = default_parameters()
integration_dim, entity_ids = lower_integral_type(Vfe.cell, "cell")
macro_cfg = dict(interface=macro_builder,
ufl_cell=Vf.ufl_cell(),
precision=parameters["precision"],
integration_dim=integration_dim,
entity_ids=entity_ids,
index_cache=index_cache,
quadrature_rule=macro_quadrature_rule)
fexpr, = fem.compile_ufl(f, **macro_cfg)
X = ufl.SpatialCoordinate(Vf.mesh())
C_a, = fem.compile_ufl(X, **macro_cfg)
detJ = ufl_utils.preprocess_expression(
abs(ufl.JacobianDeterminant(f.ufl_domain())))
macro_detJ, = fem.compile_ufl(detJ, **macro_cfg)
Vce = create_element(Vc.ufl_element())
coarse_builder = firedrake_interface.KernelBuilder("cell", "otherwise", 0,
ScalarType_c)
coarse_builder.set_coordinates(Vc.mesh())
argument_multiindices = (Vce.get_indices(), )
argument_multiindex, = argument_multiindices
return_variable, = coarse_builder.set_arguments((ufl.TestFunction(Vc), ),
argument_multiindices)
integration_dim, entity_ids = lower_integral_type(Vce.cell, "cell")
# Midpoint quadrature for jacobian on coarse cell.
quadrature_rule = make_quadrature(Vce.cell, 0)
coarse_cfg = dict(interface=coarse_builder,
ufl_cell=Vc.ufl_cell(),
precision=parameters["precision"],
integration_dim=integration_dim,
entity_ids=entity_ids,
index_cache=index_cache,
quadrature_rule=quadrature_rule)
X = ufl.SpatialCoordinate(Vc.mesh())
K = ufl_utils.preprocess_expression(ufl.JacobianInverse(Vc.mesh()))
C_0, = fem.compile_ufl(X, **coarse_cfg)
K, = fem.compile_ufl(K, **coarse_cfg)
i = gem.Index()
j = gem.Index()
C_0 = gem.Indexed(C_0, (j, ))
C_0 = gem.index_sum(C_0, quadrature_rule.point_set.indices)
C_a = gem.Indexed(C_a, (j, ))
X_a = gem.Sum(C_0, gem.Product(gem.Literal(-1), C_a))
K_ij = gem.Indexed(K, (i, j))
K_ij = gem.index_sum(K_ij, quadrature_rule.point_set.indices)
X_a = gem.index_sum(gem.Product(K_ij, X_a), (j, ))
C_0, = quadrature_rule.point_set.points
C_0 = gem.Indexed(gem.Literal(C_0), (i, ))
# fine quad points in coarse reference space.
X_a = gem.Sum(C_0, gem.Product(gem.Literal(-1), X_a))
X_a = gem.ComponentTensor(X_a, (i, ))
# Coarse basis function evaluated at fine quadrature points
phi_c = fem.fiat_to_ufl(
Vce.point_evaluation(0, X_a, (Vce.cell.get_dimension(), 0)), 0)
tensor_indices = tuple(gem.Index(extent=d) for d in f.ufl_shape)
phi_c = gem.Indexed(phi_c, argument_multiindex + tensor_indices)
fexpr = gem.Indexed(fexpr, tensor_indices)
quadrature_weight = macro_quadrature_rule.weight_expression
expr = gem.Product(gem.IndexSum(gem.Product(phi_c, fexpr), tensor_indices),
gem.Product(macro_detJ, quadrature_weight))
quadrature_indices = macro_builder.indices + macro_quadrature_rule.point_set.indices
reps = spectral.Integrals([expr], quadrature_indices,
argument_multiindices, parameters)
assignments = spectral.flatten([(return_variable, reps)], index_cache)
return_variables, expressions = zip(*assignments)
expressions = impero_utils.preprocess_gem(expressions,
**spectral.finalise_options)
assignments = list(zip(return_variables, expressions))
impero_c = impero_utils.compile_gem(assignments,
quadrature_indices +
argument_multiindex,
remove_zeros=True)
index_names = []
def name_index(index, name):
index_names.append((index, name))
if index in index_cache:
for multiindex, suffix in zip(index_cache[index],
string.ascii_lowercase):
name_multiindex(multiindex, name + suffix)
def name_multiindex(multiindex, name):
if len(multiindex) == 1:
name_index(multiindex[0], name)
else:
for i, index in enumerate(multiindex):
name_index(index, name + str(i))
name_multiindex(quadrature_indices, 'ip')
for multiindex, name in zip(argument_multiindices, ['j', 'k']):
name_multiindex(multiindex, name)
index_names.extend(zip(macro_builder.indices, ["entity"]))
body = generate_coffee(impero_c, index_names, parameters["precision"],
ScalarType_c)
retarg = ast.Decl(ScalarType_c,
ast.Symbol("R", rank=(Vce.space_dimension(), )))
local_tensor = coarse_builder.local_tensor
local_tensor.init = ast.ArrayInit(
numpy.zeros(Vce.space_dimension(), dtype=ScalarType_c))
body.children.insert(0, local_tensor)
args = [retarg] + macro_builder.kernel_args + [
macro_builder.coordinates_arg, coarse_builder.coordinates_arg
]
# Now we have the kernel that computes <f, phi_c>dx_c
# So now we need to hit it with the inverse mass matrix on dx_c
u = TrialFunction(Vc)
v = TestFunction(Vc)
expr = Tensor(ufl.inner(u, v) * ufl.dx).inv * AssembledVector(
ufl.Coefficient(Vc))
Ainv, = compile_expression(expr)
Ainv = Ainv.kinfo.kernel
A = ast.Symbol(local_tensor.sym.symbol)
R = ast.Symbol("R")
body.children.append(
ast.FunCall(Ainv.name, R, coarse_builder.coordinates_arg.sym, A))
from coffee.base import Node
assert isinstance(Ainv._code, Node)
return op2.Kernel(ast.Node([
Ainv._code,
ast.FunDecl("void",
"pyop2_kernel_injection_dg",
args,
body,
pred=["static", "inline"])
]),
name="pyop2_kernel_injection_dg",
cpp=True,
include_dirs=Ainv._include_dirs,
headers=Ainv._headers)
def component_tensor(self, o, expression, index):
index = tuple(i for i in index if i in expression.free_indices)
return gem.ComponentTensor(expression, index)
