Source code for

# Copyright (C) 2009-2020 Anders Logg, Martin Sandve Aln├Žs, Marie E. Rognes,
# Kristian B. Oelgaard, Matthew W. Scroggs, Chris Richardson, and others
# This file is part of FFCx. (
# SPDX-License-Identifier:    LGPL-3.0-or-later
"""Compiler stage 2: Code representation.

Module computes intermediate representations of forms, elements and
dofmaps. For each UFC function, we extract the data needed for code
generation at a later stage.

The representation should conform strictly to the naming and order of
functions in UFC. Thus, for code generation of the function "foo", one
should only need to use the data stored in the intermediate
representation under the key "foo".

import itertools
import logging
import numbers
import typing
import warnings

import numpy

import basix
import ufl
from ffcx import naming
from ffcx.analysis import UFLData
from ffcx.element_interface import convert_element
from import compute_integral_ir
from import (QuadratureRule,
from ufl.classes import Integral
from ufl.sorting import sorted_expr_sum

logger = logging.getLogger("ffcx")

[docs]class FormIR(typing.NamedTuple): id: int name: str signature: str rank: int num_coefficients: int num_constants: int name_from_uflfile: str function_spaces: typing.Dict[str, typing.Tuple[str, str, str, int, basix.CellType, basix.LagrangeVariant]] original_coefficient_position: typing.List[int] coefficient_names: typing.List[str] constant_names: typing.List[str] finite_elements: typing.List[str] dofmaps: typing.List[str] integral_names: typing.Dict[str, typing.List[str]] subdomain_ids: typing.Dict[str, typing.List[int]]
[docs]class CustomElementIR(typing.NamedTuple): cell_type: basix.CellType value_shape: typing.Tuple[int, ...] wcoeffs: numpy.typing.NDArray[numpy.float64] x: typing.List[typing.List[numpy.typing.NDArray[numpy.float64]]] M: typing.List[typing.List[numpy.typing.NDArray[numpy.float64]]] map_type: basix.MapType sobolev_space: basix.SobolevSpace interpolation_nderivs: int discontinuous: bool highest_complete_degree: int highest_degree: int
[docs]class ElementIR(typing.NamedTuple): id: int name: str signature: str cell_shape: str topological_dimension: int geometric_dimension: int space_dimension: int value_shape: typing.Tuple[int, ...] reference_value_shape: typing.Tuple[int, ...] degree: int family: str num_sub_elements: int block_size: int sub_elements: typing.List[str] element_type: str entity_dofs: typing.List[typing.List[typing.List[int]]] lagrange_variant: basix.LagrangeVariant dpc_variant: basix.DPCVariant basix_family: basix.ElementFamily basix_cell: basix.CellType discontinuous: bool custom_element: CustomElementIR
[docs]class DofMapIR(typing.NamedTuple): id: int name: str signature: str num_global_support_dofs: int num_element_support_dofs: int entity_dofs: typing.List[typing.List[typing.List[int]]] num_entity_dofs: typing.List[typing.List[int]] entity_closure_dofs: typing.List[typing.List[typing.List[int]]] num_entity_closure_dofs: typing.List[typing.List[int]] num_sub_dofmaps: int sub_dofmaps: typing.List[str] block_size: int
[docs]class IntegralIR(typing.NamedTuple): integral_type: str subdomain_id: typing.Union[str, typing.Tuple[int, ...], int] rank: int geometric_dimension: int topological_dimension: int entitytype: str num_facets: int num_vertices: int enabled_coefficients: typing.List[bool] element_dimensions: typing.Dict[ufl.FiniteElementBase, int] element_ids: typing.Dict[ufl.FiniteElementBase, int] tensor_shape: typing.List[int] coefficient_numbering: typing.Dict[ufl.Coefficient, int] coefficient_offsets: typing.Dict[ufl.Coefficient, int] original_constant_offsets: typing.Dict[ufl.Constant, int] options: dict cell_shape: str unique_tables: typing.Dict[str, numpy.typing.NDArray[numpy.float64]] unique_table_types: typing.Dict[str, str] integrand: typing.Dict[QuadratureRule, dict] name: str precision: int needs_facet_permutations: bool coordinate_element: str
[docs]class ExpressionIR(typing.NamedTuple): name: str element_dimensions: typing.Dict[ufl.FiniteElementBase, int] options: dict unique_tables: typing.Dict[str, numpy.typing.NDArray[numpy.float64]] unique_table_types: typing.Dict[str, str] integrand: typing.Dict[QuadratureRule, dict] coefficient_numbering: typing.Dict[ufl.Coefficient, int] coefficient_offsets: typing.Dict[ufl.Coefficient, int] integral_type: str entitytype: str tensor_shape: typing.List[int] expression_shape: typing.List[int] original_constant_offsets: typing.Dict[ufl.Constant, int] points: numpy.typing.NDArray[numpy.float64] coefficient_names: typing.List[str] constant_names: typing.List[str] needs_facet_permutations: bool function_spaces: typing.Dict[str, typing.Tuple[str, str, str, int, basix.CellType, basix.LagrangeVariant]] name_from_uflfile: str original_coefficient_positions: typing.List[int]
[docs]class DataIR(typing.NamedTuple): elements: typing.List[ElementIR] dofmaps: typing.List[DofMapIR] integrals: typing.List[IntegralIR] forms: typing.List[FormIR] expressions: typing.List[ExpressionIR]
[docs]def compute_ir(analysis: UFLData, object_names, prefix, options, visualise): """Compute intermediate representation.""" * "*")"Compiler stage 2: Computing intermediate representation of objects") * "*") # Compute object names # NOTE: This is done here for performance reasons, because repeated calls # within each IR computation would be expensive due to UFL signature computations finite_element_names = {e: naming.finite_element_name(e, prefix) for e in analysis.unique_elements} dofmap_names = {e: naming.dofmap_name(e, prefix) for e in analysis.unique_elements} integral_names = {} form_names = {} for fd_index, fd in enumerate(analysis.form_data): form_names[fd_index] = naming.form_name(fd.original_form, fd_index, prefix) for itg_index, itg_data in enumerate(fd.integral_data): integral_names[(fd_index, itg_index)] = naming.integral_name(fd.original_form, itg_data.integral_type, fd_index, itg_data.subdomain_id, prefix) ir_elements = [ _compute_element_ir(e, analysis.element_numbers, finite_element_names) for e in analysis.unique_elements ] ir_dofmaps = [ _compute_dofmap_ir(e, analysis.element_numbers, dofmap_names) for e in analysis.unique_elements ] irs = [ _compute_integral_ir(fd, i, analysis.element_numbers, integral_names, finite_element_names, options, visualise) for (i, fd) in enumerate(analysis.form_data) ] ir_integrals = list(itertools.chain(*irs)) ir_forms = [ _compute_form_ir(fd, i, prefix, form_names, integral_names, analysis.element_numbers, finite_element_names, dofmap_names, object_names) for (i, fd) in enumerate(analysis.form_data) ] ir_expressions = [_compute_expression_ir(expr, i, prefix, analysis, options, visualise, object_names, finite_element_names, dofmap_names) for i, expr in enumerate(analysis.expressions)] return DataIR(elements=ir_elements, dofmaps=ir_dofmaps, integrals=ir_integrals, forms=ir_forms, expressions=ir_expressions)
def _compute_element_ir(element, element_numbers, finite_element_names): """Compute intermediate representation of element.""""Computing IR for element {element}") element = convert_element(element) # Create basix elements cell = element.cell() # Store id ir = {"id": element_numbers[element]} ir["name"] = finite_element_names[element] # Compute data for each function ir["signature"] = repr(element) ir["cell_shape"] = ir["topological_dimension"] = cell.topological_dimension() ir["geometric_dimension"] = cell.geometric_dimension() ir["space_dimension"] = element.dim + element.num_global_support_dofs ir["element_type"] = element.ufcx_element_type ir["lagrange_variant"] = element.lagrange_variant ir["dpc_variant"] = element.dpc_variant ir["basix_family"] = element.element_family ir["basix_cell"] = element.cell_type ir["discontinuous"] = element.discontinuous ir["degree"] = ir["family"] = element.family_name ir["value_shape"] = element.value_shape() ir["reference_value_shape"] = element.reference_value_shape() ir["num_sub_elements"] = element.num_sub_elements() ir["sub_elements"] = [finite_element_names[e] for e in element.sub_elements()] ir["block_size"] = element.block_size if element.block_size > 1: element = element.sub_element ir["entity_dofs"] = element.entity_dofs if element.is_custom_element: ir["custom_element"] = _compute_custom_element_ir(element.element) else: ir["custom_element"] = None return ElementIR(**ir) def _compute_custom_element_ir(basix_element: basix.finite_element.FiniteElement): """Compute intermediate representation of a custom Basix element.""" ir = {} ir["cell_type"] = basix_element.cell_type ir["value_shape"] = basix_element.value_shape ir["wcoeffs"] = basix_element.wcoeffs ir["x"] = basix_element.x ir["M"] = basix_element.M ir["map_type"] = basix_element.map_type ir["sobolev_space"] = basix_element.sobolev_space ir["discontinuous"] = basix_element.discontinuous ir["interpolation_nderivs"] = basix_element.interpolation_nderivs ir["highest_complete_degree"] = basix_element.highest_complete_degree ir["highest_degree"] = basix_element.highest_degree return CustomElementIR(**ir) def _compute_dofmap_ir(element, element_numbers, dofmap_names): """Compute intermediate representation of dofmap.""""Computing IR for dofmap of {element}") # Create basix elements element = convert_element(element) # Store id ir = {"id": element_numbers[element]} ir["name"] = dofmap_names[element] # Compute data for each function ir["signature"] = "FFCx dofmap for " + repr(element) ir["sub_dofmaps"] = [dofmap_names[e] for e in element.sub_elements()] ir["num_sub_dofmaps"] = element.num_sub_elements() ir["block_size"] = element.block_size if element.block_size > 1: element = element.sub_element # Precompute repeatedly used items for i in element.num_entity_dofs: # FIXME: this assumes the same number of DOFs on each entity of the same dim: this # assumption will not be true for prisms and pyramids if max(i) != min(i): raise RuntimeError("Elements with different numbers of DOFs on subentities of the same dimension" " are not yet supported in FFCx.") # FIXME: This does not work for prisms and pyramids num_dofs_per_entity = [i[0] for i in element.num_entity_dofs] ir["num_entity_dofs"] = num_dofs_per_entity ir["entity_dofs"] = element.entity_dofs num_dofs_per_entity_closure = [i[0] for i in element.num_entity_closure_dofs] ir["num_entity_closure_dofs"] = num_dofs_per_entity_closure ir["entity_closure_dofs"] = element.entity_closure_dofs ir["num_global_support_dofs"] = element.num_global_support_dofs ir["num_element_support_dofs"] = element.dim return DofMapIR(**ir) def _compute_integral_ir(form_data, form_index, element_numbers, integral_names, finite_element_names, options, visualise): """Compute intermediate representation 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):"Computing IR for integral in integral group {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(), "enabled_coefficients": itg_data.enabled_coefficients, "cell_shape": cellname, "coordinate_element": finite_element_names[convert_element(itg_data.domain.ufl_coordinate_element())] } # Get element space dimensions unique_elements = element_numbers.keys() ir["element_dimensions"] = {element: element.dim + element.num_global_support_dofs for element in unique_elements} ir["element_ids"] = { element: i for i, 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"][convert_element(element)] for 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"] if scheme == "custom": points = md["quadrature_points"] weights = md["quadrature_weights"] elif scheme == "vertex": # FIXME: Could this come from basix? # 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. degree = md["quadrature_degree"] 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: degree = md["quadrature_degree"] 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"][convert_element(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=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"], options, visualise) ir.update(integral_ir) # Fetch name ir["name"] = integral_names[(form_index, itg_data_index)] irs.append(IntegralIR(**ir)) return irs def _compute_form_ir(form_data, form_id, prefix, form_names, integral_names, element_numbers, finite_element_names, dofmap_names, object_names) -> FormIR: """Compute intermediate representation of form.""""Computing IR for form {form_id}") # Store id ir = {"id": form_id} # Compute common data ir["name"] = form_names[form_id] ir["signature"] = form_data.original_form.signature() ir["rank"] = len(form_data.original_form.arguments()) ir["num_coefficients"] = len(form_data.reduced_coefficients) ir["num_constants"] = len(form_data.original_form.constants()) ir["coefficient_names"] = [object_names.get(id(obj), f"w{j}") for j, obj in enumerate(form_data.reduced_coefficients)] ir["constant_names"] = [object_names.get(id(obj), f"c{j}") for j, obj in enumerate(form_data.original_form.constants())] ir["original_coefficient_position"] = form_data.original_coefficient_positions ir["finite_elements"] = [ finite_element_names[convert_element(e)] for e in form_data.argument_elements + form_data.coefficient_elements ] ir["dofmaps"] = [ dofmap_names[convert_element(e)] for e in form_data.argument_elements + form_data.coefficient_elements ] fs = {} for function in form_data.original_form.arguments() + tuple(form_data.reduced_coefficients): name = object_names.get(id(function), str(function)) el = convert_element(convert_element(function.ufl_function_space().ufl_element())) cmap = function.ufl_function_space().ufl_domain().ufl_coordinate_element() # Default point spacing for CoordinateElement is equispaced if not isinstance(cmap, basix.ufl_wrapper._BasixElementBase) and cmap.variant() is None: cmap._sub_element._variant = "equispaced" cmap = convert_element(cmap) family = degree = fs[name] = (finite_element_names[el], dofmap_names[el], family, degree, cmap.cell_type, cmap.lagrange_variant) form_name = object_names.get(id(form_data.original_form), form_id) ir["function_spaces"] = fs ir["name_from_uflfile"] = f"form_{prefix}_{form_name}" # Store names of integrals and subdomain_ids for this form, grouped # by integral types Since form points to all integrals it contains, # it has to know their names for codegen phase ir["integral_names"] = {} ir["subdomain_ids"] = {} ufcx_integral_types = ("cell", "exterior_facet", "interior_facet") for integral_type in ufcx_integral_types: ir["subdomain_ids"][integral_type] = [] ir["integral_names"][integral_type] = [] for itg_index, itg_data in enumerate(form_data.integral_data): if (itg_data.integral_type == integral_type): if itg_data.subdomain_id == "otherwise": # UFL is using "otherwise" for default integrals # (over whole mesh) but FFCx needs integers, so # otherwise = -1 if len(ir["subdomain_ids"][integral_type]) > 0 and ir["subdomain_ids"][integral_type][0] == -1: raise ValueError("Only one default ('otherwise') integral allowed.") # Put default integral as first ir["subdomain_ids"][integral_type] = [-1] + ir["subdomain_ids"][integral_type] ir["integral_names"][integral_type] = [ integral_names[(form_id, itg_index)]] + ir["integral_names"][integral_type] elif itg_data.subdomain_id < 0: raise ValueError("Integral subdomain ID must be non-negative.") else: assert isinstance(itg_data.subdomain_id, numbers.Integral) ir["subdomain_ids"][integral_type] += [itg_data.subdomain_id] ir["integral_names"][integral_type] += [integral_names[(form_id, itg_index)]] return FormIR(**ir) def _compute_expression_ir(expression, index, prefix, analysis, options, visualise, object_names, finite_element_names, dofmap_names): """Compute intermediate representation of expression.""""Computing IR for expression {index}") # Compute representation ir = {} original_expression = (expression[2], expression[1]) ir["name"] = naming.expression_name(original_expression, prefix) original_expression = expression[2] points = expression[1] expression = expression[0] try: cell = ufl.domain.extract_unique_domain(expression).ufl_cell() except AttributeError: # This case corresponds to a spatially constant expression # without any dependencies cell = None # Prepare dimensions of all unique element in expression, including # elements for arguments, coefficients and coordinate mappings ir["element_dimensions"] = {element: element.dim + element.num_global_support_dofs for element in analysis.unique_elements} # Extract dimensions for elements of arguments only arguments = ufl.algorithms.extract_arguments(expression) argument_elements = tuple(convert_element(f.ufl_function_space().ufl_element()) for f in arguments) argument_dimensions = [ir["element_dimensions"][element] for element in argument_elements] tensor_shape = argument_dimensions ir["tensor_shape"] = tensor_shape ir["expression_shape"] = list(expression.ufl_shape) coefficients = ufl.algorithms.extract_coefficients(expression) coefficient_numbering = {} for i, coeff in enumerate(coefficients): coefficient_numbering[coeff] = i # Add coefficient numbering to IR ir["coefficient_numbering"] = coefficient_numbering original_coefficient_positions = [] original_coefficients = ufl.algorithms.extract_coefficients(original_expression) for coeff in coefficients: original_coefficient_positions.append(original_coefficients.index(coeff)) ir["coefficient_names"] = [object_names.get(id(obj), f"w{j}") for j, obj in enumerate(coefficients)] ir["constant_names"] = [object_names.get(id(obj), f"c{j}") for j, obj in enumerate(ufl.algorithms.analysis.extract_constants(expression))] fs = {} for function in tuple(original_coefficients) + tuple(arguments): name = object_names.get(id(function), str(function)) el = convert_element(function.ufl_function_space().ufl_element()) cmap = convert_element(function.ufl_function_space().ufl_domain().ufl_coordinate_element()) family = degree = fs[name] = (finite_element_names[el], dofmap_names[el], family, degree) expression_name = object_names.get(id(original_expression), index) ir["function_spaces"] = fs ir["name_from_uflfile"] = f"expression_{prefix}_{expression_name}" if len(argument_elements) > 1: raise RuntimeError("Expression with more than one Argument not implemented.") ir["original_coefficient_positions"] = original_coefficient_positions coefficient_elements = tuple(convert_element(f.ufl_element()) for f in coefficients) offsets = {} _offset = 0 for i, el in enumerate(coefficient_elements): offsets[coefficients[i]] = _offset _offset += ir["element_dimensions"][convert_element(el)] # Copy offsets also into IR ir["coefficient_offsets"] = offsets ir["integral_type"] = "expression" ir["entitytype"] = "cell" # Build offsets for Constants original_constant_offsets = {} _offset = 0 for constant in ufl.algorithms.analysis.extract_constants(expression): original_constant_offsets[constant] = _offset _offset += numpy.product(constant.ufl_shape, dtype=int) ir["original_constant_offsets"] = original_constant_offsets ir["points"] = points weights = numpy.array([1.0] * points.shape[0]) rule = QuadratureRule(points, weights) integrands = {rule: expression} if cell is None: assert len(ir["original_coefficient_positions"]) == 0 and len(ir["original_constant_offsets"]) == 0 expression_ir = compute_integral_ir(cell, ir["integral_type"], ir["entitytype"], integrands, tensor_shape, options, visualise) ir.update(expression_ir) return ExpressionIR(**ir)