"""Types for representing symbolic expressions for geometric quantities."""
# Copyright (C) 2008-2016 Martin Sandve Alnæs
#
# This file is part of UFL (https://www.fenicsproject.org)
#
# SPDX-License-Identifier: LGPL-3.0-or-later
from ufl.core.terminal import Terminal
from ufl.core.ufl_type import ufl_type
from ufl.domain import as_domain, extract_unique_domain
from ufl.sobolevspace import H1
"""
Possible coordinate bootstrapping:
Xf = Xf[q]
FacetCoordinate = quadrature point on facet (ds,dS)
X = X[q]
CellCoordinate = quadrature point on cell (dx)
x = x[q]
SpatialCoordinate = quadrature point from input array (dc)
Jacobians of mappings between coordinates:
Jcf = dX/dXf = grad_Xf X(Xf)
CellFacetJacobian
Jxc = dx/dX = grad_X x(X)
Jacobian
Jxf = dx/dXf = grad_Xf x(Xf) = Jxc Jcf = dx/dX dX/dXf = grad_X x(X) grad_Xf X(Xf)
FacetJacobian = Jacobian * CellFacetJacobian
Possible computation of X from Xf:
X = Jcf Xf + X0f
CellCoordinate = CellFacetJacobian * FacetCoordinate + CellFacetOrigin
Possible computation of x from X:
x = f(X)
SpatialCoordinate = sum_k xdofs_k * xphi_k(X)
x = Jxc X + x0
SpatialCoordinate = Jacobian * CellCoordinate + CellOrigin
Possible computation of x from Xf:
x = x(X(Xf))
x = Jxf Xf + x0f
SpatialCoordinate = FacetJacobian * FacetCoordinate + FacetOrigin
Inverse relations:
X = K * (x - x0)
CellCoordinate = JacobianInverse * (SpatialCoordinate - CellOrigin)
Xf = FK * (x - x0f)
FacetCoordinate = FacetJacobianInverse * (SpatialCoordinate - FacetOrigin)
Xf = CFK * (X - X0f)
FacetCoordinate = CellFacetJacobianInverse * (CellCoordinate - CellFacetOrigin)
"""
# --- Expression node types
[docs]@ufl_type(is_abstract=True)
class GeometricQuantity(Terminal):
"""Geometric quantity."""
__slots__ = ("_domain",)
def __init__(self, domain):
"""Initialise."""
Terminal.__init__(self)
self._domain = as_domain(domain)
[docs] def ufl_domains(self):
"""Get the UFL domains."""
return (self._domain,)
[docs] def is_cellwise_constant(self):
"""Return whether this expression is spatially constant over each cell."""
# NB! Geometric quantities are piecewise constant by
# default. Override if needed.
return True
# NB! Geometric quantities are scalar by default. Override if
# needed.
ufl_shape = ()
def _ufl_signature_data_(self, renumbering):
"""Signature data of geometric quantities depend on the domain numbering."""
return (self._ufl_class_.__name__,) + self._domain._ufl_signature_data_(renumbering)
def __str__(self):
"""Format as a string."""
return self._ufl_class_.name
def __repr__(self):
"""Representation."""
r = "%s(%s)" % (self._ufl_class_.__name__, repr(self._domain))
return r
def _ufl_compute_hash_(self):
"""UFL compute hash."""
return hash((type(self).__name__,) + self._domain._ufl_hash_data_())
def __eq__(self, other):
"""Check equality."""
return isinstance(other, self._ufl_class_) and other._domain == self._domain
[docs]@ufl_type(is_abstract=True)
class GeometricCellQuantity(GeometricQuantity):
"""Geometric cell quantity."""
__slots__ = ()
[docs]@ufl_type(is_abstract=True)
class GeometricFacetQuantity(GeometricQuantity):
"""Geometric facet quantity."""
__slots__ = ()
# --- Coordinate represented in different coordinate systems
[docs]@ufl_type()
class SpatialCoordinate(GeometricCellQuantity):
"""The coordinate in a domain.
In the context of expression integration,
represents the domain coordinate of each quadrature point.
In the context of expression evaluation in a point,
represents the value of that point.
"""
__slots__ = ()
name = "x"
@property
def ufl_shape(self):
"""Return the number of coordinates defined (i.e. the geometric dimension of the domain)."""
g = self._domain.geometric_dimension()
return (g,)
[docs] def is_cellwise_constant(self):
"""Return whether this expression is spatially constant over each cell."""
# Only case this is true is if the domain is a vertex cell.
t = self._domain.topological_dimension()
return t == 0
[docs] def evaluate(self, x, mapping, component, index_values):
"""Return the value of the coordinate."""
if component == ():
if isinstance(x, (tuple, list)):
return float(x[0])
else:
return float(x)
else:
return float(x[component[0]])
[docs] def count(self):
"""Count."""
# FIXME: Hack to make SpatialCoordinate behave like a coefficient.
# When calling `derivative`, the count is used to sort over.
return -1
[docs]@ufl_type()
class CellCoordinate(GeometricCellQuantity):
"""The coordinate in a reference cell.
In the context of expression integration,
represents the reference cell coordinate of each quadrature point.
In the context of expression evaluation in a point in a cell,
represents that point in the reference coordinate system of the cell.
"""
__slots__ = ()
name = "X"
@property
def ufl_shape(self):
"""Get the UFL shape."""
t = self._domain.topological_dimension()
return (t,)
[docs] def is_cellwise_constant(self):
"""Return whether this expression is spatially constant over each cell."""
# Only case this is true is if the domain is a vertex cell.
t = self._domain.topological_dimension()
return t == 0
[docs]@ufl_type()
class FacetCoordinate(GeometricFacetQuantity):
"""The coordinate in a reference cell of a facet.
In the context of expression integration over a facet,
represents the reference facet coordinate of each quadrature point.
In the context of expression evaluation in a point on a facet,
represents that point in the reference coordinate system of the facet.
"""
__slots__ = ()
name = "Xf"
def __init__(self, domain):
"""Initialise."""
GeometricFacetQuantity.__init__(self, domain)
t = self._domain.topological_dimension()
if t < 2:
raise ValueError("FacetCoordinate is only defined for topological dimensions >= 2.")
@property
def ufl_shape(self):
"""Get the UFL shape."""
t = self._domain.topological_dimension()
return (t - 1,)
[docs] def is_cellwise_constant(self):
"""Return whether this expression is spatially constant over each cell."""
# Only case this is true is if the domain is an interval cell
# (with a vertex facet).
t = self._domain.topological_dimension()
return t <= 1
# --- Origin of coordinate systems in larger coordinate systems
[docs]@ufl_type()
class CellOrigin(GeometricCellQuantity):
"""The spatial coordinate corresponding to origin of a reference cell."""
__slots__ = ()
name = "x0"
@property
def ufl_shape(self):
"""Get the UFL shape."""
g = self._domain.geometric_dimension()
return (g,)
[docs] def is_cellwise_constant(self):
"""Return whether this expression is spatially constant over each cell."""
return True
[docs]@ufl_type()
class FacetOrigin(GeometricFacetQuantity):
"""The spatial coordinate corresponding to origin of a reference facet."""
__slots__ = ()
name = "x0f"
@property
def ufl_shape(self):
"""Get the UFL shape."""
g = self._domain.geometric_dimension()
return (g,)
[docs]@ufl_type()
class CellFacetOrigin(GeometricFacetQuantity):
"""The reference cell coordinate corresponding to origin of a reference facet."""
__slots__ = ()
name = "X0f"
@property
def ufl_shape(self):
"""Get the UFL shape."""
t = self._domain.topological_dimension()
return (t,)
# --- Jacobians of mappings between coordinate systems
[docs]@ufl_type()
class Jacobian(GeometricCellQuantity):
r"""The Jacobian of the mapping from reference cell to spatial coordinates.
.. math:: J_{ij} = \\frac{dx_i}{dX_j}
"""
__slots__ = ()
name = "J"
@property
def ufl_shape(self):
"""Return the number of coordinates defined (i.e. the geometric dimension of the domain)."""
g = self._domain.geometric_dimension()
t = self._domain.topological_dimension()
return (g, t)
[docs] def is_cellwise_constant(self):
"""Return whether this expression is spatially constant over each cell."""
# Only true for a piecewise linear coordinate field in simplex cells
return self._domain.is_piecewise_linear_simplex_domain()
[docs]@ufl_type()
class FacetJacobian(GeometricFacetQuantity):
"""The Jacobian of the mapping from reference facet to spatial coordinates.
FJ_ij = dx_i/dXf_j
The FacetJacobian is the product of the Jacobian and CellFacetJacobian:
FJ = dx/dXf = dx/dX dX/dXf = J * CFJ
"""
__slots__ = ()
name = "FJ"
def __init__(self, domain):
"""Initialise."""
GeometricFacetQuantity.__init__(self, domain)
t = self._domain.topological_dimension()
if t < 2:
raise ValueError("FacetJacobian is only defined for topological dimensions >= 2.")
@property
def ufl_shape(self):
"""Get the UFL shape."""
g = self._domain.geometric_dimension()
t = self._domain.topological_dimension()
return (g, t - 1)
[docs] def is_cellwise_constant(self):
"""Return whether this expression is spatially constant over each cell."""
# Only true for a piecewise linear coordinate field in simplex
# cells
return self._domain.is_piecewise_linear_simplex_domain()
[docs]@ufl_type()
class CellFacetJacobian(GeometricFacetQuantity): # dX/dXf
"""The Jacobian of the mapping from reference facet to reference cell coordinates.
CFJ_ij = dX_i/dXf_j
"""
__slots__ = ()
name = "CFJ"
def __init__(self, domain):
"""Initialise."""
GeometricFacetQuantity.__init__(self, domain)
t = self._domain.topological_dimension()
if t < 2:
raise ValueError("CellFacetJacobian is only defined for topological dimensions >= 2.")
@property
def ufl_shape(self):
"""Get the UFL shape."""
t = self._domain.topological_dimension()
return (t, t - 1)
[docs] def is_cellwise_constant(self):
"""Return whether this expression is spatially constant over each cell."""
# This is always a constant mapping between two reference
# coordinate systems.
return True
[docs]@ufl_type()
class ReferenceCellEdgeVectors(GeometricCellQuantity):
"""The vectors between reference cell vertices for each edge in cell."""
__slots__ = ()
name = "RCEV"
def __init__(self, domain):
"""Initialise."""
GeometricCellQuantity.__init__(self, domain)
t = self._domain.topological_dimension()
if t < 2:
raise ValueError("CellEdgeVectors is only defined for topological dimensions >= 2.")
@property
def ufl_shape(self):
"""Get the UFL shape."""
cell = extract_unique_domain(self).ufl_cell()
ne = cell.num_edges()
t = cell.topological_dimension()
return (ne, t)
[docs] def is_cellwise_constant(self):
"""Return whether this expression is spatially constant over each cell."""
# This is always constant for a given cell type
return True
[docs]@ufl_type()
class ReferenceFacetEdgeVectors(GeometricFacetQuantity):
"""The vectors between reference cell vertices for each edge in current facet."""
__slots__ = ()
name = "RFEV"
def __init__(self, domain):
"""Initialise."""
GeometricFacetQuantity.__init__(self, domain)
t = self._domain.topological_dimension()
if t < 3:
raise ValueError("FacetEdgeVectors is only defined for topological dimensions >= 3.")
@property
def ufl_shape(self):
"""Get the UFL shape."""
cell = extract_unique_domain(self).ufl_cell()
facet_types = cell.facet_types()
# Raise exception for cells with more than one facet type e.g. prisms
if len(facet_types) > 1:
raise Exception(f"Cell type {cell} not supported.")
nfe = facet_types[0].num_edges()
t = cell.topological_dimension()
return (nfe, t)
[docs] def is_cellwise_constant(self):
"""Return whether this expression is spatially constant over each cell."""
# This is always constant for a given cell type
return True
[docs]@ufl_type()
class CellVertices(GeometricCellQuantity):
"""Physical cell vertices."""
__slots__ = ()
name = "CV"
def __init__(self, domain):
"""Initialise."""
GeometricCellQuantity.__init__(self, domain)
@property
def ufl_shape(self):
"""Get the UFL shape."""
domain = extract_unique_domain(self)
cell = domain.ufl_cell()
nv = cell.num_vertices()
g = domain.geometric_dimension()
return (nv, g)
[docs] def is_cellwise_constant(self):
"""Return whether this expression is spatially constant over each cell."""
# This is always constant for a given cell type
return True
[docs]@ufl_type()
class CellEdgeVectors(GeometricCellQuantity):
"""The vectors between physical cell vertices for each edge in cell."""
__slots__ = ()
name = "CEV"
def __init__(self, domain):
"""Initialise."""
GeometricCellQuantity.__init__(self, domain)
t = self._domain.topological_dimension()
if t < 2:
raise ValueError("CellEdgeVectors is only defined for topological dimensions >= 2.")
@property
def ufl_shape(self):
"""Get the UFL shape."""
domain = extract_unique_domain(self)
cell = domain.ufl_cell()
ne = cell.num_edges()
g = domain.geometric_dimension()
return (ne, g)
[docs] def is_cellwise_constant(self):
"""Return whether this expression is spatially constant over each cell."""
# This is always constant for a given cell type
return True
[docs]@ufl_type()
class FacetEdgeVectors(GeometricFacetQuantity):
"""The vectors between physical cell vertices for each edge in current facet."""
__slots__ = ()
name = "FEV"
def __init__(self, domain):
"""Initialise."""
GeometricFacetQuantity.__init__(self, domain)
t = self._domain.topological_dimension()
if t < 3:
raise ValueError("FacetEdgeVectors is only defined for topological dimensions >= 3.")
@property
def ufl_shape(self):
"""Get the UFL shape."""
domain = extract_unique_domain(self)
cell = domain.ufl_cell()
facet_types = cell.facet_types()
# Raise exception for cells with more than one facet type e.g. prisms
if len(facet_types) > 1:
raise Exception(f"Cell type {cell} not supported.")
nfe = facet_types[0].num_edges()
g = domain.geometric_dimension()
return (nfe, g)
[docs] def is_cellwise_constant(self):
"""Return whether this expression is spatially constant over each cell."""
# This is always constant for a given cell type
return True
# --- Determinants (signed or pseudo) of geometry mapping Jacobians
[docs]@ufl_type()
class JacobianDeterminant(GeometricCellQuantity):
"""The determinant of the Jacobian.
Represents the signed determinant of a square Jacobian or the
pseudo-determinant of a non-square Jacobian.
"""
__slots__ = ()
name = "detJ"
[docs] def is_cellwise_constant(self):
"""Return whether this expression is spatially constant over each cell."""
# Only true for a piecewise linear coordinate field in simplex
# cells
return self._domain.is_piecewise_linear_simplex_domain()
[docs]@ufl_type()
class FacetJacobianDeterminant(GeometricFacetQuantity):
"""The pseudo-determinant of the FacetJacobian."""
__slots__ = ()
name = "detFJ"
[docs] def is_cellwise_constant(self):
"""Return whether this expression is spatially constant over each cell."""
# Only true for a piecewise linear coordinate field in simplex cells
return self._domain.is_piecewise_linear_simplex_domain()
[docs]@ufl_type()
class CellFacetJacobianDeterminant(GeometricFacetQuantity):
"""The pseudo-determinant of the CellFacetJacobian."""
__slots__ = ()
name = "detCFJ"
[docs] def is_cellwise_constant(self):
"""Return whether this expression is spatially constant over each cell."""
# Only true for a piecewise linear coordinate field in simplex
# cells
return self._domain.is_piecewise_linear_simplex_domain()
# --- Inverses (signed or pseudo) of geometry mapping Jacobians
[docs]@ufl_type()
class JacobianInverse(GeometricCellQuantity):
"""The inverse of the Jacobian.
Represents the inverse of a square Jacobian or the pseudo-inverse of
a non-square Jacobian.
"""
__slots__ = ()
name = "K"
@property
def ufl_shape(self):
"""Return the number of coordinates defined (i.e. the geometric dimension of the domain)."""
g = self._domain.geometric_dimension()
t = self._domain.topological_dimension()
return (t, g)
[docs] def is_cellwise_constant(self):
"""Return whether this expression is spatially constant over each cell."""
# Only true for a piecewise linear coordinate field in simplex
# cells
return self._domain.is_piecewise_linear_simplex_domain()
[docs]@ufl_type()
class FacetJacobianInverse(GeometricFacetQuantity):
"""The pseudo-inverse of the FacetJacobian."""
__slots__ = ()
name = "FK"
def __init__(self, domain):
"""Initialise."""
GeometricFacetQuantity.__init__(self, domain)
t = self._domain.topological_dimension()
if t < 2:
raise ValueError(
"FacetJacobianInverse is only defined for topological dimensions >= 2."
)
@property
def ufl_shape(self):
"""Get the UFL shape."""
g = self._domain.geometric_dimension()
t = self._domain.topological_dimension()
return (t - 1, g)
[docs] def is_cellwise_constant(self):
"""Return whether this expression is spatially constant over each cell."""
# Only true for a piecewise linear coordinate field in simplex
# cells
return self._domain.is_piecewise_linear_simplex_domain()
[docs]@ufl_type()
class CellFacetJacobianInverse(GeometricFacetQuantity):
"""The pseudo-inverse of the CellFacetJacobian."""
__slots__ = ()
name = "CFK"
def __init__(self, domain):
"""Initialise."""
GeometricFacetQuantity.__init__(self, domain)
t = self._domain.topological_dimension()
if t < 2:
raise ValueError(
"CellFacetJacobianInverse is only defined for topological dimensions >= 2."
)
@property
def ufl_shape(self):
"""Get the UFL shape."""
t = self._domain.topological_dimension()
return (t - 1, t)
[docs] def is_cellwise_constant(self):
"""Return whether this expression is spatially constant over each cell."""
# Only true for a piecewise linear coordinate field in simplex cells
return self._domain.is_piecewise_linear_simplex_domain()
# --- Types representing normal or tangent vectors
[docs]@ufl_type()
class FacetNormal(GeometricFacetQuantity):
"""The outwards pointing normal vector of the current facet."""
__slots__ = ()
name = "n"
@property
def ufl_shape(self):
"""Return the number of coordinates defined (i.e. the geometric dimension of the domain)."""
g = self._domain.geometric_dimension()
return (g,)
[docs] def is_cellwise_constant(self):
"""Return whether this expression is spatially constant over each cell."""
# For product cells, this is only true for some but not all
# facets. Seems like too much work to fix right now. Only
# true for a piecewise linear coordinate field with simplex
# _facets_.
ce = self._domain.ufl_coordinate_element()
is_piecewise_linear = ce.embedded_superdegree <= 1 and ce in H1
return is_piecewise_linear and self._domain.ufl_cell().has_simplex_facets()
[docs]@ufl_type()
class CellNormal(GeometricCellQuantity):
"""The upwards pointing normal vector of the current manifold cell."""
__slots__ = ()
name = "cell_normal"
@property
def ufl_shape(self):
"""Return the number of coordinates defined (i.e. the geometric dimension of the domain)."""
g = self._domain.geometric_dimension()
# t = self._domain.topological_dimension()
# return (g-t,g) # TODO: Should it be CellNormals? For interval in 3D we have two!
return (g,)
[docs] def is_cellwise_constant(self):
"""Return whether this expression is spatially constant over each cell."""
# Only true for a piecewise linear coordinate field in simplex cells
return self._domain.is_piecewise_linear_simplex_domain()
[docs]@ufl_type()
class ReferenceNormal(GeometricFacetQuantity):
"""The outwards pointing normal vector of the current facet on the reference cell."""
__slots__ = ()
name = "reference_normal"
@property
def ufl_shape(self):
"""Get the UFL shape."""
t = self._domain.topological_dimension()
return (t,)
# --- Types representing measures of the cell and entities of the cell,
# typically used for stabilisation terms
# TODO: Clean up this set of types? Document!
[docs]@ufl_type()
class ReferenceCellVolume(GeometricCellQuantity):
"""The volume of the reference cell."""
__slots__ = ()
name = "reference_cell_volume"
[docs]@ufl_type()
class ReferenceFacetVolume(GeometricFacetQuantity):
"""The volume of the reference cell of the current facet."""
__slots__ = ()
name = "reference_facet_volume"
[docs]@ufl_type()
class CellVolume(GeometricCellQuantity):
"""The volume of the cell."""
__slots__ = ()
name = "volume"
[docs]@ufl_type()
class Circumradius(GeometricCellQuantity):
"""The circumradius of the cell."""
__slots__ = ()
name = "circumradius"
[docs]@ufl_type()
class CellDiameter(GeometricCellQuantity):
"""The diameter of the cell, i.e., maximal distance of two points in the cell."""
__slots__ = ()
name = "diameter"
[docs]@ufl_type()
class FacetArea(GeometricFacetQuantity): # FIXME: Should this be allowed for interval domain?
"""The area of the facet."""
__slots__ = ()
name = "facetarea"
[docs]@ufl_type()
class MinCellEdgeLength(GeometricCellQuantity):
"""The minimum edge length of the cell."""
__slots__ = ()
name = "mincelledgelength"
[docs]@ufl_type()
class MaxCellEdgeLength(GeometricCellQuantity):
"""The maximum edge length of the cell."""
__slots__ = ()
name = "maxcelledgelength"
[docs]@ufl_type()
class MinFacetEdgeLength(GeometricFacetQuantity):
"""The minimum edge length of the facet."""
__slots__ = ()
name = "minfacetedgelength"
[docs]@ufl_type()
class MaxFacetEdgeLength(GeometricFacetQuantity):
"""The maximum edge length of the facet."""
__slots__ = ()
name = "maxfacetedgelength"
# --- Types representing other stuff
[docs]@ufl_type()
class CellOrientation(GeometricCellQuantity):
"""The orientation (+1/-1) of the current cell.
For non-manifold cells (tdim == gdim), this equals the sign
of the Jacobian determinant, i.e. +1 if the physical cell is
oriented the same way as the reference cell and -1 otherwise.
For manifold cells of tdim==gdim-1 this is input data belonging
to the mesh, used to distinguish between the sides of the manifold.
"""
__slots__ = ()
name = "cell_orientation"
[docs]@ufl_type()
class FacetOrientation(GeometricFacetQuantity):
"""The orientation (+1/-1) of the current facet relative to the reference cell."""
__slots__ = ()
name = "facet_orientation"
# This doesn't quite fit anywhere. Make a special set of symbolic
# terminal types instead?
[docs]@ufl_type()
class QuadratureWeight(GeometricQuantity):
"""The current quadrature weight.
Only used inside a quadrature context.
"""
__slots__ = ()
name = "weight"
[docs] def is_cellwise_constant(self):
"""Return whether this expression is spatially constant over each cell."""
# The weight usually varies with the quadrature points
return False