def _reduce_facet_edge_length(self, o, reduction_op):
if self._preserve_types[o._ufl_typecode_]:
return o
domain = o.ufl_domain()
if domain.ufl_cell().topological_dimension() < 3:
error(
"Facet edge lengths only make sense for topological dimension >= 3."
)
elif not domain.ufl_coordinate_element().degree() == 1:
# Don't lower bendy cells, instead leave it to form compiler
warning(
"Only know how to compute facet edge lengths of P1 or Q1 cell."
)
return o
else:
# P1 tetrahedron or Q1 hexahedron
edges = FacetEdgeVectors(domain)
num_edges = edges.ufl_shape[0]
j = Index()
elen2 = [edges[e, j] * edges[e, j] for e in range(num_edges)]
return sqrt(reduce(reduction_op, elen2))
def cell_normal(self, o):
if self._preserve_types[o._ufl_typecode_]:
return o
domain = o.ufl_domain()
gdim = domain.geometric_dimension()
tdim = domain.topological_dimension()
if tdim == gdim - 1: # n-manifold embedded in n-1 space
i = Index()
J = self.jacobian(Jacobian(domain))
if tdim == 2:
# Surface in 3D
t0 = as_vector(J[i, 0], i)
t1 = as_vector(J[i, 1], i)
cell_normal = cross_expr(t0, t1)
elif tdim == 1:
# Line in 2D (cell normal is 'up' for a line pointing
# to the 'right')
cell_normal = as_vector((-J[1, 0], J[0, 0]))
else:
error("Cell normal not implemented for tdim %d, gdim %d" %
(tdim, gdim))
# Return normalized vector, sign corrected by cell
# orientation
co = CellOrientation(domain)
return co * cell_normal / sqrt(cell_normal[i] * cell_normal[i])
else:
error("What do you want cell normal in gdim={0}, tdim={1} to be?".
format(gdim, tdim))
def dot(self, o, a, b):
ai = indices(len(a.ufl_shape) - 1)
bi = indices(len(b.ufl_shape) - 1)
k = (Index(), )
# Creates a single IndexSum over a Product
s = a[ai + k] * b[k + bi]
return as_tensor(s, ai + bi)
def zero(self, o):
fi = o.ufl_free_indices
fid = o.ufl_index_dimensions
mapped_fi = tuple(self.index(Index(count=i)) for i in fi)
paired_fid = [(mapped_fi[pos], fid[pos]) for pos, a in enumerate(fi)]
new_fi, new_fid = zip(*tuple(sorted(paired_fid)))
return Zero(o.ufl_shape, new_fi, new_fid)
def nabla_grad(self, o, a):
sh = a.ufl_shape
if sh == ():
return Grad(a)
else:
j = Index()
ii = tuple(indices(len(sh)))
return as_tensor(a[ii].dx(j), (j,) + ii)
def facet_normal(self, o):
if self._preserve_types[o._ufl_typecode_]:
return o
domain = o.ufl_domain()
tdim = domain.topological_dimension()
if tdim == 1:
# Special-case 1D (possibly immersed), for which we say
# that n is just in the direction of J.
J = self.jacobian(Jacobian(domain)) # dx/dX
ndir = J[:, 0]
gdim = domain.geometric_dimension()
if gdim == 1:
nlen = abs(ndir[0])
else:
i = Index()
nlen = sqrt(ndir[i] * ndir[i])
rn = ReferenceNormal(domain) # +/- 1.0 here
n = rn[0] * ndir / nlen
r = n
else:
# Recall that the covariant Piola transform u -> J^(-T)*u
# preserves tangential components. The normal vector is
# characterised by having zero tangential component in
# reference and physical space.
Jinv = self.jacobian_inverse(JacobianInverse(domain))
i, j = indices(2)
rn = ReferenceNormal(domain)
# compute signed, unnormalised normal; note transpose
ndir = as_vector(Jinv[j, i] * rn[j], i)
# normalise
i = Index()
n = ndir / sqrt(ndir[i] * ndir[i])
r = n
if r.ufl_shape != o.ufl_shape:
error("Inconsistent dimensions (in=%d, out=%d)." %
(o.ufl_shape[0], r.ufl_shape[0]))
return r
def index(self, o):
if isinstance(o, FixedIndex):
return o
else:
c = o._count
i = self.index_map.get(c)
if i is None:
i = Index(count=len(self.index_map))
self.index_map[c] = i
return i
def altenative_dot(self, o, a, b): # TODO: Test this
ash = a.ufl_shape
bsh = b.ufl_shape
ai = indices(len(ash) - 1)
bi = indices(len(bsh) - 1)
# Simplification for tensors where the dot-sum dimension has
# length 1
if ash[-1] == 1:
k = (FixedIndex(0),)
else:
k = (Index(),)
# Potentially creates a single IndexSum over a Product
s = a[ai+k]*b[k+bi]
return as_tensor(s, ai+bi)
def alternative_inner(self, o, a, b): # TODO: Test this
ash = a.ufl_shape
bsh = b.ufl_shape
if ash != bsh:
error("Nonmatching shapes.")
# Simplification for tensors with one or more dimensions of
# length 1
ii = []
zi = FixedIndex(0)
for n in ash:
if n == 1:
ii.append(zi)
else:
ii.append(Index())
ii = tuple(ii)
# ii = indices(len(a.ufl_shape))
# Potentially creates multiple IndexSums over a Product
s = a[ii]*Conj(b[ii])
return s
def cell_diameter(self, o):
if self._preserve_types[o._ufl_typecode_]:
return o
domain = o.ufl_domain()
if not domain.ufl_coordinate_element().degree() in {1, (1, 1)}:
# Don't lower bendy cells, instead leave it to form compiler
warning("Only know how to compute cell diameter of P1 or Q1 cell.")
return o
elif domain.is_piecewise_linear_simplex_domain():
# Simplices
return self.max_cell_edge_length(MaxCellEdgeLength(domain))
else:
# Q1 cells, maximal distance between any two vertices
verts = CellVertices(domain)
verts = [verts[v, ...] for v in range(verts.ufl_shape[0])]
j = Index()
elen2 = (real((v0 - v1)[j] * conj((v0 - v1)[j])) for v0, v1 in combinations(verts, 2))
return real(sqrt(reduce(max_value, elen2)))
def _reduce_cell_edge_length(self, o, reduction_op):
if self._preserve_types[o._ufl_typecode_]:
return o
domain = o.ufl_domain()
if not domain.ufl_coordinate_element().degree() == 1:
# Don't lower bendy cells, instead leave it to form compiler
warning("Only know how to compute cell edge lengths of P1 or Q1 cell.")
return o
elif domain.ufl_cell().cellname() == "interval":
# Interval optimization, square root not needed
return self.cell_volume(CellVolume(domain))
else:
# Other P1 or Q1 cells
edges = CellEdgeVectors(domain)
num_edges = edges.ufl_shape[0]
j = Index()
elen2 = [real(edges[e, j] * conj(edges[e, j])) for e in range(num_edges)]
return real(sqrt(reduce(reduction_op, elen2)))
def create_slice_indices(component, shape, fi):
all_indices = []
slice_indices = []
repeated_indices = []
free_indices = []
for ind in component:
if isinstance(ind, Index):
all_indices.append(ind)
if ind.count() in fi or ind in free_indices:
repeated_indices.append(ind)
free_indices.append(ind)
elif isinstance(ind, FixedIndex):
if int(ind) >= shape[len(all_indices)]:
error("Index out of bounds.")
all_indices.append(ind)
elif isinstance(ind, int):
if int(ind) >= shape[len(all_indices)]:
error("Index out of bounds.")
all_indices.append(FixedIndex(ind))
elif isinstance(ind, slice):
if ind != slice(None):
error("Only full slices (:) allowed.")
i = Index()
slice_indices.append(i)
all_indices.append(i)
elif ind == Ellipsis:
er = len(shape) - len(component) + 1
ii = indices(er)
slice_indices.extend(ii)
all_indices.extend(ii)
else:
error("Not expecting {0}.".format(ind))
if len(all_indices) != len(shape):
error("Component and shape length don't match.")
return tuple(all_indices), tuple(slice_indices), tuple(repeated_indices)
def circumradius(self, o):
if self._preserve_types[o._ufl_typecode_]:
return o
domain = o.ufl_domain()
if not domain.is_piecewise_linear_simplex_domain():
error("Circumradius only makes sense for affine simplex cells")
cellname = domain.ufl_cell().cellname()
cellvolume = self.cell_volume(CellVolume(domain))
if cellname == "interval":
# Optimization for square interval; no square root needed
return 0.5 * cellvolume
# Compute lengths of cell edges
edges = CellEdgeVectors(domain)
num_edges = edges.ufl_shape[0]
j = Index()
elen = [sqrt(edges[e, j] * edges[e, j]) for e in range(num_edges)]
if cellname == "triangle":
return (elen[0] * elen[1] * elen[2]) / (4.0 * cellvolume)
elif cellname == "tetrahedron":
# la, lb, lc = lengths of the sides of an intermediate triangle
# NOTE: Is here some hidden numbering assumption?
la = elen[3] * elen[2]
lb = elen[4] * elen[1]
lc = elen[5] * elen[0]
# p = perimeter
p = (la + lb + lc)
# s = semiperimeter
s = p / 2
# area of intermediate triangle with Herons formula
triangle_area = sqrt(s * (s - la) * (s - lb) * (s - lc))
return triangle_area / (6.0 * cellvolume)
def analyse_key(ii, rank):
"""Takes something the user might input as an index tuple
inside [], which could include complete slices (:) and
ellipsis (...), and returns tuples of actual UFL index objects.
The return value is a tuple (indices, axis_indices),
each being a tuple of IndexBase instances.
The return value 'indices' corresponds to all
input objects of these types:
- Index
- FixedIndex
- int => Wrapped in FixedIndex
The return value 'axis_indices' corresponds to all
input objects of these types:
- Complete slice (:) => Replaced by a single new index
- Ellipsis (...) => Replaced by multiple new indices
"""
# Wrap in tuple
if not isinstance(ii, (tuple, MultiIndex)):
ii = (ii, )
else:
# Flatten nested tuples, happens with f[...,ii] where ii is a
# tuple of indices
jj = []
for j in ii:
if isinstance(j, (tuple, MultiIndex)):
jj.extend(j)
else:
jj.append(j)
ii = tuple(jj)
# Convert all indices to Index or FixedIndex objects. If there is
# an ellipsis, split the indices into before and after.
axis_indices = set()
pre = []
post = []
indexlist = pre
for i in ii:
if i == Ellipsis:
# Switch from pre to post list when an ellipsis is
# encountered
if indexlist is not pre:
error("Found duplicate ellipsis.")
indexlist = post
else:
# Convert index to a proper type
if isinstance(i, numbers.Integral):
idx = FixedIndex(i)
elif isinstance(i, IndexBase):
idx = i
elif isinstance(i, slice):
if i == slice(None):
idx = Index()
axis_indices.add(idx)
else:
# TODO: Use ListTensor to support partial slices?
error(
"Partial slices not implemented, only complete slices like [:]"
)
else:
error("Can't convert this object to index: %s" % (i, ))
# Store index in pre or post list
indexlist.append(idx)
# Handle ellipsis as a number of complete slices, that is create a
# number of new axis indices
num_axis = rank - len(pre) - len(post)
if indexlist is post:
ellipsis_indices = indices(num_axis)
axis_indices.update(ellipsis_indices)
else:
ellipsis_indices = ()
# Construct final tuples to return
all_indices = tuple(chain(pre, ellipsis_indices, post))
axis_indices = tuple(i for i in all_indices if i in axis_indices)
return all_indices, axis_indices
def trace(self, o, A):
i = Index()
return A[i, i]
def nabla_div(self, o, a):
i = Index()
return a[i, ...].dx(i)
def div(self, o, a):
i = Index()
return a[..., i].dx(i)
def _mult(a, b):
# Discover repeated indices, which results in index sums
afi = a.ufl_free_indices
bfi = b.ufl_free_indices
afid = a.ufl_index_dimensions
bfid = b.ufl_index_dimensions
fi, fid, ri, rid = merge_overlapping_indices(afi, afid, bfi, bfid)
# Pick out valid non-scalar products here (dot products):
# - matrix-matrix (A*B, M*grad(u)) => A . B
# - matrix-vector (A*v) => A . v
s1, s2 = a.ufl_shape, b.ufl_shape
r1, r2 = len(s1), len(s2)
if r1 == 0 and r2 == 0:
# Create scalar product
p = Product(a, b)
ti = ()
elif r1 == 0 or r2 == 0:
# Scalar - tensor product
if r2 == 0:
a, b = b, a
# Check for zero, simplifying early if possible
if isinstance(a, Zero) or isinstance(b, Zero):
shape = s1 or s2
return Zero(shape, fi, fid)
# Repeated indices are allowed, like in:
# v[i]*M[i,:]
# Apply product to scalar components
ti = indices(len(b.ufl_shape))
p = Product(a, b[ti])
elif r1 == 2 and r2 in (1, 2): # Matrix-matrix or matrix-vector
if ri:
error("Not expecting repeated indices in non-scalar product.")
# Check for zero, simplifying early if possible
if isinstance(a, Zero) or isinstance(b, Zero):
shape = s1[:-1] + s2[1:]
return Zero(shape, fi, fid)
# Return dot product in index notation
ai = indices(len(a.ufl_shape) - 1)
bi = indices(len(b.ufl_shape) - 1)
k = indices(1)
p = a[ai + k] * b[k + bi]
ti = ai + bi
else:
error("Invalid ranks {0} and {1} in product.".format(r1, r2))
# TODO: I think applying as_tensor after index sums results in
# cleaner expression graphs.
# Wrap as tensor again
if ti:
p = as_tensor(p, ti)
# If any repeated indices were found, apply implicit summation
# over those
for i in ri:
mi = MultiIndex((Index(count=i), ))
p = IndexSum(p, mi)
return p