def generate_tabulate_dmats(L, dofs_data):
"""Tabulate the derivatives of the polynomial base"""
alignas = 32
# Emit code for the dmats we've actually used
dmats_code = [
L.Comment("Tables of derivatives of the polynomial base (transpose).")
]
dmats_names = []
all_matrices = []
for idof, dof_data in enumerate(dofs_data):
# Get derivative matrices (coefficients) of basis functions, computed by FIAT at compile time.
derivative_matrices = dof_data["dmats"]
num_mats = len(derivative_matrices)
num_members = dof_data["num_expansion_members"]
# Generate tables for each spatial direction.
matrix = numpy.zeros((num_mats, num_members, num_members))
for i, dmat in enumerate(derivative_matrices):
# Extract derivatives for current direction
# (take transpose, FIAT_NEW PolynomialSet.tabulate()).
matrix[i, ...] = numpy.transpose(dmat)
# TODO: Use precision from parameters here
from ffc.ir.uflacs.elementtables import clamp_table_small_numbers
matrix = clamp_table_small_numbers(matrix)
# O(n^2) matrix matching...
name = None
for oldname, oldmatrix in all_matrices:
if matrix.shape == oldmatrix.shape and numpy.allclose(
matrix, oldmatrix):
name = oldname
break
if name is None:
# Define variable name for coefficients for this dof
name = L.Symbol("dmats%d" % (idof, ))
all_matrices.append((name, matrix))
# Declare new dmats table with unique values
decl = L.ArrayDecl("static const double",
name, (num_mats, num_members, num_members),
values=matrix,
alignas=alignas)
dmats_code.append(decl)
# Append name for each dof
dmats_names.append(name)
return dmats_names, dmats_code
def compute_midpoint_geometry(L, ir):
# Dimensions
gdim = ir.geometric_dimension
tdim = ir.topological_dimension
num_dofs = ir.num_scalar_coordinate_element_dofs
# Tables of coordinate basis function values and derivatives at
# X=0 and X=midpoint available through ir. This is useful in
# several geometry functions.
tables = ir.tables
# Check the table shapes against our expectations
xm_table = tables["xm"]
Jm_table = tables["Jm"]
assert xm_table.shape == (num_dofs, )
assert Jm_table.shape == (tdim, num_dofs)
# TODO: Use epsilon parameter here?
# TODO: Move to a more 'neutral' utility file
from ffc.ir.uflacs.elementtables import clamp_table_small_numbers
xm_table = clamp_table_small_numbers(xm_table)
Jm_table = clamp_table_small_numbers(Jm_table)
# Table symbols
phi_Xm = L.Symbol("phi_Xm")
dphi_Xm = L.Symbol("dphi_Xm")
# Table declarations
table_decls = [
L.ArrayDecl("const double",
phi_Xm,
sizes=xm_table.shape,
values=xm_table),
L.ArrayDecl("const double",
dphi_Xm,
sizes=Jm_table.shape,
values=Jm_table),
]
# Output geometry
x = L.Symbol("x")
J = L.Symbol("J")
# Input cell data
coordinate_dofs = L.Symbol("coordinate_dofs")
coordinate_dofs = L.FlattenedArray(coordinate_dofs, dims=(num_dofs, gdim))
Jf = L.FlattenedArray(J, dims=(gdim, tdim))
i = L.Symbol("i")
j = L.Symbol("j")
d = L.Symbol("d")
iz = L.Symbol("l") # Index for zeroing arrays
# Initialise arrays to zero
init_array = [
L.ForRange(iz,
0,
gdim * tdim,
index_type=index_type,
body=L.Assign(Jf.array[iz], 0.0))
]
xm_code = [
L.Comment("Compute x"),
L.ForRange(i,
0,
gdim,
index_type=index_type,
body=[
L.Assign(x[i], 0.0),
L.ForRange(d,
0,
num_dofs,
index_type=index_type,
body=L.AssignAdd(
x[i], coordinate_dofs[d, i] * phi_Xm[d]))
]),
]
Jm_code = [
L.Comment("Compute J"),
L.ForRanges((i, 0, gdim), (j, 0, tdim), (d, 0, num_dofs),
index_type=index_type,
body=L.AssignAdd(Jf[i, j],
coordinate_dofs[d, i] * dphi_Xm[j, d])),
]
# Reuse functions for detJ and K
code = table_decls + init_array + xm_code + Jm_code
return code
def _compute_reference_coordinates_affine(L, ir, output_all=False):
# Class name
classname = ir.classname
# Dimensions
gdim = ir.geometric_dimension
tdim = ir.topological_dimension
num_points = L.Symbol("num_points")
# Number of dofs for a scalar component
num_dofs = ir.num_scalar_coordinate_element_dofs
# Loop indices
ip = L.Symbol("ip") # point
i = L.Symbol("i") # gdim
j = L.Symbol("j") # tdim
k = L.Symbol("k") # sum iteration
iz = L.Symbol("l") # zero X array
# Output geometry
X = L.FlattenedArray(L.Symbol("X"), dims=(num_points, tdim))
# if output_all, this is also output:
Jsym = L.Symbol("J")
detJsym = L.Symbol("detJ")
Ksym = L.Symbol("K")
# Input geometry
x = L.FlattenedArray(L.Symbol("x"), dims=(num_points, gdim))
# Input cell data
coordinate_dofs = L.FlattenedArray(L.Symbol("coordinate_dofs"),
dims=(num_dofs, gdim))
cell_orientation = L.Symbol("cell_orientation")
init_input = [
L.ForRange(iz,
0,
num_points * tdim,
index_type=index_type,
body=L.Assign(X.array[iz], 0.0))
]
if output_all:
decls = []
else:
decls = [
L.ArrayDecl("double", Jsym, sizes=(gdim * tdim, )),
L.ArrayDecl("double", detJsym, sizes=(1, )),
L.ArrayDecl("double", Ksym, sizes=(tdim * gdim, )),
]
# Tables of coordinate basis function values and derivatives at X=0
# and X=midpoint available through ir. This is useful in several
# geometry functions.
tables = ir.tables
# Check the table shapes against our expectations
x_table = tables["x0"]
J_table = tables["J0"]
assert x_table.shape == (num_dofs, )
assert J_table.shape == (tdim, num_dofs)
# TODO: Use epsilon parameter here?
# TODO: Move to a more 'neutral' utility file
from ffc.ir.uflacs.elementtables import clamp_table_small_numbers
x_table = clamp_table_small_numbers(x_table)
J_table = clamp_table_small_numbers(J_table)
# Table symbols
phi_X0 = L.Symbol("phi_X0")
dphi_X0 = L.Symbol("dphi_X0")
# Table declarations
table_decls = [
L.ArrayDecl("const double",
phi_X0,
sizes=x_table.shape,
values=x_table),
L.ArrayDecl("const double",
dphi_X0,
sizes=J_table.shape,
values=J_table),
]
# Compute x0 = x(X=0) (optimized by precomputing basis at X=0)
x0 = L.Symbol("x0")
compute_x0 = [
L.ArrayDecl("double", x0, sizes=(gdim, ), values=0),
L.ForRanges((i, 0, gdim), (k, 0, num_dofs),
index_type=index_type,
body=L.AssignAdd(x0[i],
coordinate_dofs[k, i] * phi_X0[k])),
]
# For more convenient indexing
J = L.FlattenedArray(Jsym, dims=(gdim, tdim))
K = L.FlattenedArray(Ksym, dims=(tdim, gdim))
# Compute J = J(X=0) (optimized by precomputing basis at X=0)
compute_J0 = [
L.ForRanges((i, 0, gdim), (j, 0, tdim),
index_type=index_type,
body=[
L.Assign(J[i, j], 0.0),
L.ForRange(k,
0,
num_dofs,
index_type=index_type,
body=L.AssignAdd(
J[i, j],
coordinate_dofs[k, i] * dphi_X0[j, k]))
]),
]
# Compute K = inv(J) (and intermediate value det(J))
compute_K0 = [
L.Call("compute_jacobian_determinants_{}".format(classname),
(detJsym, 1, Jsym, cell_orientation)),
L.Call("compute_jacobian_inverses_{}".format(classname),
(Ksym, 1, Jsym, detJsym)),
]
# Compute X = K0*(x-x0) for each physical point x
compute_X = [
L.ForRanges((ip, 0, num_points), (j, 0, tdim), (i, 0, gdim),
index_type=index_type,
body=L.AssignAdd(X[ip, j], K[j, i] * (x[ip, i] - x0[i])))
]
# Stitch it together
code = init_input + table_decls + decls + compute_x0 + compute_J0 + compute_K0 + compute_X
return code
def build_uflacs_ir(cell, integral_type, entitytype, integrands, tensor_shape,
quadrature_rules, parameters):
# The intermediate representation dict we're building and returning
# here
ir = {}
# Extract uflacs specific optimization and code generation
# parameters
p = parse_uflacs_optimization_parameters(parameters, integral_type)
# Pass on parameters for consumption in code generation
ir["params"] = p
# Shared unique tables for all quadrature loops
ir["unique_tables"] = {}
ir["unique_table_types"] = {}
# Shared piecewise expr_ir for all quadrature loops
ir["piecewise_ir"] = {"factorization": None,
"modified_arguments": [],
"preintegrated_blocks": {},
"premultiplied_blocks": {},
"preintegrated_contributions": collections.defaultdict(list),
"block_contributions": collections.defaultdict(list)}
# { num_points: expr_ir for one integrand }
ir["varying_irs"] = {"factorization": None}
# Whether we expect the quadrature weight to be applied or not (in
# some cases it's just set to 1 in ufl integral scaling)
tdim = cell.topological_dimension()
expect_weight = (integral_type not in point_integral_types and (entitytype == "cell" or (
entitytype == "facet" and tdim > 1) or (integral_type in custom_integral_types)))
# Analyse each num_points/integrand separately
assert isinstance(integrands, dict)
all_num_points = sorted(integrands.keys())
cases = [(num_points, [integrands[num_points]]) for num_points in all_num_points]
ir["all_num_points"] = all_num_points
for num_points, expressions in cases:
assert len(expressions) == 1
expression = expressions[0]
# Rebalance order of nested terminal modifiers
expression = balance_modifiers(expression)
# Remove QuadratureWeight terminals from expression and replace with 1.0
expression = replace_quadratureweight(expression)
# Build initial scalar list-based graph representation
S = build_scalar_graph(expression)
S_targets = [i for i, v in S.nodes.items() if v.get('target', False)]
assert len(S_targets) == 1
S_target = S_targets[0]
# Build terminal_data from V here before factorization. Then we
# can use it to derive table properties for all modified
# terminals, and then use that to rebuild the scalar graph more
# efficiently before argument factorization. We can build
# terminal_data again after factorization if that's necessary.
initial_terminals = {i: analyse_modified_terminal(v['expression'])
for i, v in S.nodes.items()
if is_modified_terminal(v['expression'])}
unique_tables, unique_table_types, unique_table_num_dofs, mt_unique_table_reference = build_optimized_tables(
num_points,
quadrature_rules,
cell,
integral_type,
entitytype,
initial_terminals.values(),
ir["unique_tables"],
p["enable_table_zero_compression"],
rtol=p["table_rtol"],
atol=p["table_atol"])
# If there are any 'zero' tables, replace symbolically and rebuild graph
if 'zeros' in unique_table_types.values():
for i, mt in initial_terminals.items():
# Set modified terminals with zero tables to zero
tr = mt_unique_table_reference.get(mt)
if tr is not None and tr.ttype == "zeros":
S.nodes[i]['expression'] = ufl.as_ufl(0.0)
# Propagate expression changes using dependency list
for i, v in S.nodes.items():
deps = [S.nodes[j]['expression'] for j in S.out_edges[i]]
if deps:
v['expression'] = v['expression']._ufl_expr_reconstruct_(*deps)
# Rebuild scalar target expressions and graph (this may be
# overkill and possible to optimize away if it turns out to be
# costly)
expression = S.nodes[S_target]['expression']
# Rebuild scalar list-based graph representation
S = build_scalar_graph(expression)
# Output diagnostic graph as pdf
if parameters['visualise']:
visualise(S, 'S.pdf')
# Compute factorization of arguments
rank = len(tensor_shape)
F = compute_argument_factorization(S, rank)
# Get the 'target' nodes that are factors of arguments, and insert in dict
FV_targets = [i for i, v in F.nodes.items() if v.get('target', False)]
argument_factorization = {}
for i in FV_targets:
for w in F.nodes[i]['target']:
argument_factorization[w] = i
# Get list of indices in F which are the arguments (should be at start)
argkeys = set()
for w in argument_factorization:
argkeys = argkeys | set(w)
argkeys = list(argkeys)
# Output diagnostic graph as pdf
if parameters['visualise']:
visualise(F, 'F.pdf')
# Build set of modified_terminals for each mt factorized vertex in F
# and attach tables, if appropriate
for i, v in F.nodes.items():
expr = v['expression']
if is_modified_terminal(expr):
mt = analyse_modified_terminal(expr)
F.nodes[i]['mt'] = mt
tr = mt_unique_table_reference.get(mt)
if tr is not None:
F.nodes[i]['tr'] = tr
# Attach 'status' to each node: 'inactive', 'piecewise' or 'varying'
analyse_dependencies(F, mt_unique_table_reference)
# Save the factorisation graph to the piecewise IR
ir["piecewise_ir"]["factorization"] = F
ir["piecewise_ir"]["modified_arguments"] = [F.nodes[i]['mt']
for i in argkeys]
# Loop over factorization terms
block_contributions = collections.defaultdict(list)
for ma_indices, fi in sorted(argument_factorization.items()):
# Get a bunch of information about this term
assert rank == len(ma_indices)
trs = tuple(F.nodes[ai]['tr'] for ai in ma_indices)
unames = tuple(tr.name for tr in trs)
ttypes = tuple(tr.ttype for tr in trs)
assert not any(tt == "zeros" for tt in ttypes)
blockmap = tuple(tr.dofmap for tr in trs)
block_is_uniform = all(tr.is_uniform for tr in trs)
# Collect relevant restrictions to identify blocks correctly
# in interior facet integrals
block_restrictions = []
for i, ai in enumerate(ma_indices):
if trs[i].is_uniform:
r = None
else:
r = F.nodes[ai]['mt'].restriction
block_restrictions.append(r)
block_restrictions = tuple(block_restrictions)
factor_is_piecewise = F.nodes[fi]['status'] == 'piecewise'
# TODO: Add separate block modes for quadrature
# Both arguments in quadrature elements
"""
for iq
fw = f*w
#for i
# for j
# B[i,j] = fw*U[i]*V[j] = 0 if i != iq or j != iq
BQ[iq] = B[iq,iq] = fw
for (iq)
A[iq+offset0, iq+offset1] = BQ[iq]
"""
# One argument in quadrature element
"""
for iq
fw[iq] = f*w
#for i
# for j
# B[i,j] = fw*UQ[i]*V[j] = 0 if i != iq
for j
BQ[iq,j] = fw[iq]*V[iq,j]
for (iq) for (j)
A[iq+offset, j+offset] = BQ[iq,j]
"""
# Decide how to handle code generation for this block
if p["enable_preintegration"] and (factor_is_piecewise and rank > 0
and "quadrature" not in ttypes):
# - Piecewise factor is an absolute prerequisite
# - Could work for rank 0 as well but currently doesn't
# - Haven't considered how quadrature elements work out
block_mode = "preintegrated"
elif p["enable_premultiplication"] and (rank > 0 and all(tt in piecewise_ttypes
for tt in ttypes)):
# Integrate functional in quadloop, scale block after
# quadloop
block_mode = "premultiplied"
elif p["enable_sum_factorization"]:
if (rank == 2 and any(tt in piecewise_ttypes for tt in ttypes)):
# Partial computation in quadloop of f*u[i], compute
# (f*u[i])*v[i] outside quadloop, (or with u,v
# swapped)
block_mode = "partial"
else:
# Full runtime integration of f*u[i]*v[j], can still
# do partial computation in quadloop of f*u[i] but
# must compute (f*u[i])*v[i] as well inside
# quadloop. (or with u,v swapped)
block_mode = "full"
else:
# Use full runtime integration with nothing fancy going
# on
block_mode = "safe"
# Carry out decision
if block_mode == "preintegrated":
# Add to contributions:
# P = sum_q weight*u*v; preintegrated here
# B[...] = f * P[...]; generated after quadloop
# A[blockmap] += B[...]; generated after quadloop
cache = ir["piecewise_ir"]["preintegrated_blocks"]
block_is_transposed = False
pname = cache.get(unames)
# Reuse transpose to save memory
if p["enable_block_transpose_reuse"] and pname is None and len(unames) == 2:
pname = cache.get((unames[1], unames[0]))
if pname is not None:
# Cache hit on transpose
block_is_transposed = True
if pname is None:
# Cache miss, precompute block
weights = quadrature_rules[num_points][1]
if integral_type == "interior_facet":
ptable = integrate_block_interior_facets(
weights, unames, ttypes, unique_tables, unique_table_num_dofs)
else:
ptable = integrate_block(weights, unames, ttypes, unique_tables,
unique_table_num_dofs)
ptable = clamp_table_small_numbers(
ptable, rtol=p["table_rtol"], atol=p["table_atol"])
pname = "PI%d" % (len(cache, ))
cache[unames] = pname
unique_tables[pname] = ptable
unique_table_types[pname] = "preintegrated"
assert factor_is_piecewise
block_unames = (pname, )
blockdata = block_data_t(
block_mode, ttypes, fi, factor_is_piecewise, block_unames,
block_restrictions, block_is_transposed, block_is_uniform, pname,
None, None)
block_is_piecewise = True
elif block_mode == "premultiplied":
# Add to contributions:
# P = u*v; computed here
# FI = sum_q weight * f; generated inside quadloop
# B[...] = FI * P[...]; generated after quadloop
# A[blockmap] += B[...]; generated after quadloop
cache = ir["piecewise_ir"]["premultiplied_blocks"]
block_is_transposed = False
pname = cache.get(unames)
# Reuse transpose to save memory
if p["enable_block_transpose_reuse"] and pname is None and len(unames) == 2:
pname = cache.get((unames[1], unames[0]))
if pname is not None:
# Cache hit on transpose
block_is_transposed = True
if pname is None:
# Cache miss, precompute block
if integral_type == "interior_facet":
ptable = multiply_block_interior_facets(0, unames, ttypes, unique_tables,
unique_table_num_dofs)
else:
ptable = multiply_block(0, unames, ttypes, unique_tables,
unique_table_num_dofs)
pname = "PM%d" % (len(cache, ))
cache[unames] = pname
unique_tables[pname] = ptable
unique_table_types[pname] = "premultiplied"
block_unames = (pname, )
blockdata = block_data_t(
block_mode, ttypes, fi, factor_is_piecewise, block_unames,
block_restrictions, block_is_transposed, block_is_uniform, pname, None, None)
block_is_piecewise = False
# elif block_mode == "scaled":
# # TODO: Add mode, block is piecewise but choose not to be premultiplied
# # Add to contributions:
# # FI = sum_q weight * f; generated inside quadloop
# # B[...] = FI * u * v; generated after quadloop
# # A[blockmap] += B[...]; generated after quadloop
# raise NotImplementedError("scaled block mode not implemented.")
# # (probably need mostly the same data as
# # premultiplied, except no P table name or values)
# block_is_piecewise = False
elif block_mode in ("partial", "full", "safe"):
block_is_piecewise = factor_is_piecewise and not expect_weight
ma_data = []
for i, ma in enumerate(ma_indices):
if not trs[i].is_piecewise:
block_is_piecewise = False
ma_data.append(ma_data_t(ma, trs[i]))
block_is_transposed = False # FIXME: Handle transposes for these block types
if block_mode == "partial":
# Add to contributions:
# P[i] = sum_q weight * f * u[i]; generated inside quadloop
# B[i,j] = P[i] * v[j]; generated after quadloop (where v is the piecewise ma)
# A[blockmap] += B[...]; generated after quadloop
# Find first piecewise index TODO: Is last better? just reverse range here
for i in range(rank):
if trs[i].is_piecewise:
piecewise_ma_index = i
break
assert rank == 2
not_piecewise_ma_index = 1 - piecewise_ma_index
block_unames = (unames[not_piecewise_ma_index], )
blockdata = block_data_t(block_mode, ttypes, fi,
factor_is_piecewise, block_unames,
block_restrictions, block_is_transposed,
None, None, tuple(ma_data), piecewise_ma_index)
elif block_mode in ("full", "safe"):
# Add to contributions:
# B[i] = sum_q weight * f * u[i] * v[j]; generated inside quadloop
# A[blockmap] += B[i]; generated after quadloop
block_unames = unames
blockdata = block_data_t(block_mode, ttypes, fi,
factor_is_piecewise, block_unames,
block_restrictions, block_is_transposed,
None, None, tuple(ma_data), None)
else:
raise RuntimeError("Invalid block_mode %s" % (block_mode, ))
if block_is_piecewise:
# Insert in piecewise expr_ir
ir["piecewise_ir"]["block_contributions"][blockmap].append(blockdata)
else:
# Insert in varying expr_ir for this quadrature loop
block_contributions[blockmap].append(blockdata)
# Figure out which table names are referenced in unstructured
# partition
active_table_names = set()
for i, v in F.nodes.items():
tr = v.get('tr')
if tr is not None and F.nodes[i]['status'] != 'inactive':
active_table_names.add(tr.name)
# Figure out which table names are referenced in blocks
for blockmap, contributions in itertools.chain(
block_contributions.items(), ir["piecewise_ir"]["block_contributions"].items()):
for blockdata in contributions:
if blockdata.block_mode in ("preintegrated", "premultiplied"):
active_table_names.add(blockdata.name)
elif blockdata.block_mode in ("partial", "full", "safe"):
for mad in blockdata.ma_data:
active_table_names.add(mad.tabledata.name)
# Record all table types before dropping tables
ir["unique_table_types"].update(unique_table_types)
# Drop tables not referenced from modified terminals
# and tables of zeros and ones
unused_ttypes = ("zeros", "ones", "quadrature")
keep_table_names = set()
for name in active_table_names:
ttype = ir["unique_table_types"][name]
if ttype not in unused_ttypes:
if name in unique_tables:
keep_table_names.add(name)
unique_tables = {name: unique_tables[name] for name in keep_table_names}
# Add to global set of all tables
for name, table in unique_tables.items():
tbl = ir["unique_tables"].get(name)
if tbl is not None and not numpy.allclose(
tbl, table, rtol=p["table_rtol"], atol=p["table_atol"]):
raise RuntimeError("Table values mismatch with same name.")
ir["unique_tables"].update(unique_tables)
# Analyse active terminals to check what we'll need to generate code for
active_mts = []
for i, v in F.nodes.items():
mt = v.get('mt', False)
if mt and F.nodes[i]['status'] != 'inactive':
active_mts.append(mt)
# Figure out if we need to access CellCoordinate to avoid
# generating quadrature point table otherwise
if integral_type == "cell":
need_points = any(isinstance(mt.terminal, CellCoordinate) for mt in active_mts)
elif integral_type in facet_integral_types:
need_points = any(isinstance(mt.terminal, FacetCoordinate) for mt in active_mts)
elif integral_type in custom_integral_types:
need_points = True # TODO: Always?
else:
need_points = False
# Figure out if we need to access QuadratureWeight to avoid
# generating quadrature point table otherwise need_weights =
# any(isinstance(mt.terminal, QuadratureWeight) for mt in
# active_mts)
# Count blocks of each mode
block_modes = collections.defaultdict(int)
for blockmap, contributions in block_contributions.items():
for blockdata in contributions:
block_modes[blockdata.block_mode] += 1
# Debug output
summary = "\n".join(
" {}\t{}".format(count, mode) for mode, count in sorted(block_modes.items()))
logger.debug("Blocks of each mode: {}".format(summary))
# If there are any blocks other than preintegrated we need weights
if expect_weight and any(mode != "preintegrated" for mode in block_modes):
need_weights = True
elif integral_type in custom_integral_types:
need_weights = True # TODO: Always?
else:
need_weights = False
# Build IR dict for the given expressions
# Store final ir for this num_points
ir["varying_irs"][num_points] = {"factorization": F,
"modified_arguments": [F.nodes[i]['mt'] for i in argkeys],
"block_contributions": block_contributions,
"need_points": need_points,
"need_weights": need_weights}
return ir
