def generic_pseudo_inverse_expr(A):
"""Compute the Penrose-Moore pseudo-inverse of A: (A.T*A)^-1 * A.T."""
i, j, k = indices(3)
ATA = as_tensor(A[k, i] * A[k, j], (i, j))
ATAinv = inverse_expr(ATA)
q, r, s = indices(3)
return as_tensor(ATAinv[r, q] * A[s, q], (r, s))
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 generic_pseudo_inverse_expr(A):
"""Compute the Penrose-Moore pseudo-inverse of A: (A.T*A)^-1 * A.T."""
i, j, k = indices(3)
ATA = as_tensor(A[k, i] * A[k, j], (i, j))
ATAinv = inverse_expr(ATA)
q, r, s = indices(3)
return as_tensor(ATAinv[r, q] * A[s, q], (r, s))
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 circumradius(self, o):
if self._preserve_types[o._ufl_typecode_]:
return o
domain = o.ufl_domain()
if not domain.is_piecewise_linear_simplex_domain():
# Don't lower for non-affine cells, instead leave it to
# form compiler
warning(
"Only know how to compute the circumradius of an affine cell.")
return o
cellname = domain.ufl_cell().cellname()
cellvolume = self.cell_volume(CellVolume(domain))
if cellname == "interval":
r = 0.5 * cellvolume
elif cellname == "triangle":
J = self.jacobian(Jacobian(domain))
trev = CellEdgeVectors(domain)
num_edges = 3
i, j, k = indices(3)
elen = [
sqrt((J[i, j] * trev[edge, j]) * (J[i, k] * trev[edge, k]))
for edge in range(num_edges)
]
r = (elen[0] * elen[1] * elen[2]) / (4.0 * cellvolume)
elif cellname == "tetrahedron":
J = self.jacobian(Jacobian(domain))
trev = CellEdgeVectors(domain)
num_edges = 6
i, j, k = indices(3)
elen = [
sqrt((J[i, j] * trev[edge, j]) * (J[i, k] * trev[edge, k]))
for edge in range(num_edges)
]
# elen[3] = length of edge 3
# la, lb, lc = lengths of the sides of an intermediate triangle
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))
r = triangle_area / (6.0 * cellvolume)
else:
error("Unhandled cell type %s." % cellname)
return r
def jacobian_inverse(self, o):
# grad(K) == K_ji rgrad(K)_rj
if is_cellwise_constant(o):
return self.independent_terminal(o)
if not o._ufl_is_terminal_:
error("ReferenceValue can only wrap a terminal")
r = indices(len(o.ufl_shape))
i, j = indices(2)
Do = as_tensor(o[j, i] * ReferenceGrad(o)[r + (j, )], r + (i, ))
return Do
def jacobian_inverse(self, o):
# grad(K) == K_ji rgrad(K)_rj
if is_cellwise_constant(o):
return self.independent_terminal(o)
if not o._ufl_is_terminal_:
error("ReferenceValue can only wrap a terminal")
r = indices(len(o.ufl_shape))
i, j = indices(2)
Do = as_tensor(o[j, i]*ReferenceGrad(o)[r + (j,)], r + (i,))
return Do
def reference_value(self, o):
# grad(o) == grad(rv(f)) -> K_ji*rgrad(rv(f))_rj
f = o.ufl_operands[0]
if not f._ufl_is_terminal_:
error("ReferenceValue can only wrap a terminal")
domain = f.ufl_domain()
K = JacobianInverse(domain)
r = indices(len(o.ufl_shape))
i, j = indices(2)
Do = as_tensor(K[j, i] * ReferenceGrad(o)[r + (j, )], r + (i, ))
return Do
def reference_value(self, o):
# grad(o) == grad(rv(f)) -> K_ji*rgrad(rv(f))_rj
f = o.ufl_operands[0]
if not f._ufl_is_terminal_:
error("ReferenceValue can only wrap a terminal")
domain = f.ufl_domain()
K = JacobianInverse(domain)
r = indices(len(o.ufl_shape))
i, j = indices(2)
Do = as_tensor(K[j, i]*ReferenceGrad(o)[r + (j,)], r + (i,))
return Do
def apply_known_single_pullback(r, element):
"""Apply pullback with given mapping.
:arg r: Expression wrapped in ReferenceValue
:arg element: The element defining the mapping
"""
# Need to pass in r rather than the physical space thing, because
# the latter may be a ListTensor or similar, rather than a
# Coefficient/Argument (in the case of mixed elements, see below
# in apply_single_function_pullbacks), to which we cannot apply ReferenceValue
mapping = element.mapping()
domain = r.ufl_domain()
if mapping == "physical":
return r
elif mapping == "identity":
return r
elif mapping == "contravariant Piola":
J = Jacobian(domain)
detJ = JacobianDeterminant(J)
transform = (1.0 / detJ) * J
# Apply transform "row-wise" to TensorElement(PiolaMapped, ...)
*k, i, j = indices(len(r.ufl_shape) + 1)
kj = (*k, j)
f = as_tensor(transform[i, j] * r[kj], (*k, i))
return f
elif mapping == "covariant Piola":
K = JacobianInverse(domain)
# Apply transform "row-wise" to TensorElement(PiolaMapped, ...)
*k, i, j = indices(len(r.ufl_shape) + 1)
kj = (*k, j)
f = as_tensor(K[j, i] * r[kj], (*k, i))
return f
elif mapping == "L2 Piola":
detJ = JacobianDeterminant(domain)
return r / detJ
elif mapping == "double contravariant Piola":
J = Jacobian(domain)
detJ = JacobianDeterminant(J)
transform = (1.0 / detJ) * J
# Apply transform "row-wise" to TensorElement(PiolaMapped, ...)
*k, i, j, m, n = indices(len(r.ufl_shape) + 2)
kmn = (*k, m, n)
f = as_tensor((1.0 / detJ)**2 * J[i, m] * r[kmn] * J[j, n], (*k, i, j))
return f
elif mapping == "double covariant Piola":
K = JacobianInverse(domain)
# Apply transform "row-wise" to TensorElement(PiolaMapped, ...)
*k, i, j, m, n = indices(len(r.ufl_shape) + 2)
kmn = (*k, m, n)
f = as_tensor(K[m, i] * r[kmn] * K[n, j], (*k, i, j))
return f
else:
error("Should never be reached!")
def reference_grad(self, o):
# grad(o) == grad(rgrad(rv(f))) -> K_ji*rgrad(rgrad(rv(f)))_rj
f = o.ufl_operands[0]
valid_operand = f._ufl_is_in_reference_frame_ or isinstance(f, (JacobianInverse, SpatialCoordinate))
if not valid_operand:
error("ReferenceGrad can only wrap a reference frame type!")
domain = f.ufl_domain()
K = JacobianInverse(domain)
r = indices(len(o.ufl_shape))
i, j = indices(2)
Do = as_tensor(K[j, i]*ReferenceGrad(o)[r + (j,)], r + (i,))
return Do
def reference_grad(self, o):
# grad(o) == grad(rgrad(rv(f))) -> K_ji*rgrad(rgrad(rv(f)))_rj
f = o.ufl_operands[0]
valid_operand = f._ufl_is_in_reference_frame_ or isinstance(f, (JacobianInverse, SpatialCoordinate))
if not valid_operand:
error("ReferenceGrad can only wrap a reference frame type!")
domain = f.ufl_domain()
K = JacobianInverse(domain)
r = indices(len(o.ufl_shape))
i, j = indices(2)
Do = as_tensor(K[j, i]*ReferenceGrad(o)[r + (j,)], r + (i,))
return Do
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 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 reference_value(self, o):
# grad(o) == grad(rv(f)) -> K_ji*rgrad(rv(f))_rj
f = o.ufl_operands[0]
if f.ufl_element().mapping() == "physical":
# TODO: Do we need to be more careful for immersed things?
return ReferenceGrad(o)
if not f._ufl_is_terminal_:
error("ReferenceValue can only wrap a terminal")
domain = f.ufl_domain()
K = JacobianInverse(domain)
r = indices(len(o.ufl_shape))
i, j = indices(2)
Do = as_tensor(K[j, i] * ReferenceGrad(o)[r + (j, )], r + (i, ))
return Do
def indexed(self, o, Ap,
ii): # TODO: (Partially) duplicated in nesting rules
# Propagate zeros
if isinstance(Ap, Zero):
return self.independent_operator(o)
# Untangle as_tensor(C[kk], jj)[ii] -> C[ll] to simplify
# resulting expression
if isinstance(Ap, ComponentTensor):
B, jj = Ap.ufl_operands
if isinstance(B, Indexed):
C, kk = B.ufl_operands
kk = list(kk)
if all(j in kk for j in jj):
rep = dict(zip(jj, ii))
Cind = [rep.get(k, k) for k in kk]
expr = Indexed(C, MultiIndex(tuple(Cind)))
assert expr.ufl_free_indices == o.ufl_free_indices
assert expr.ufl_shape == o.ufl_shape
return expr
# Otherwise a more generic approach
r = len(Ap.ufl_shape) - len(ii)
if r:
kk = indices(r)
op = Indexed(Ap, MultiIndex(ii.indices() + kk))
op = as_tensor(op, kk)
else:
op = Indexed(Ap, ii)
return op
def indexed(self, o, Ap,
ii): # TODO: (Partially) duplicated in generic rules
# Reuse if untouched
if Ap is o.ufl_operands[0]:
return o
# Untangle as_tensor(C[kk], jj)[ii] -> C[ll] to simplify
# resulting expression
if isinstance(Ap, ComponentTensor):
B, jj = Ap.ufl_operands
if isinstance(B, Indexed):
C, kk = B.ufl_operands
kk = list(kk)
if all(j in kk for j in jj):
Cind = list(kk)
for i, j in zip(ii, jj):
Cind[kk.index(j)] = i
return Indexed(C, MultiIndex(tuple(Cind)))
# Otherwise a more generic approach
r = len(Ap.ufl_shape) - len(ii)
if r:
kk = indices(r)
op = Indexed(Ap, MultiIndex(ii.indices() + kk))
op = as_tensor(op, kk)
else:
op = Indexed(Ap, ii)
return op
def _make_identity(self, sh):
"Create a higher order identity tensor to represent dv/dv."
res = None
if sh == ():
# Scalar dv/dv is scalar
return FloatValue(1.0)
elif len(sh) == 1:
# Vector v makes dv/dv the identity matrix
return Identity(sh[0])
else:
# TODO: Add a type for this higher order identity?
# II[i0,i1,i2,j0,j1,j2] = 1 if all((i0==j0, i1==j1, i2==j2)) else 0
# Tensor v makes dv/dv some kind of higher rank identity tensor
ind1 = ()
ind2 = ()
for d in sh:
i, j = indices(2)
dij = Identity(d)[i, j]
if res is None:
res = dij
else:
res *= dij
ind1 += (i, )
ind2 += (j, )
fp = as_tensor(res, ind1 + ind2)
return fp
def max_facet_edge_length(self, o):
if self._preserve_types[o._ufl_typecode_]:
return o
domain = o.ufl_domain()
if not domain.is_piecewise_linear_simplex_domain():
# Don't lower for non-affine cells, instead leave it to
# form compiler
warning(
"Only know how to compute the max_facet_edge_length of an affine cell."
)
return o
cellname = domain.ufl_cell().cellname()
if cellname == "triangle":
return self.facet_area(FacetArea(domain))
elif cellname == "tetrahedron":
J = self.jacobian(Jacobian(domain))
trev = FacetEdgeVectors(domain)
num_edges = 3
i, j, k = indices(3)
elen = [
sqrt((J[i, j] * trev[edge, j]) * (J[i, k] * trev[edge, k]))
for edge in range(num_edges)
]
return max_value(elen[0], max_value(elen[1], elen[2]))
else:
error("Unhandled cell type %s." % cellname)
def _make_identity(self, sh):
"Create a higher order identity tensor to represent dv/dv."
res = None
if sh == ():
# Scalar dv/dv is scalar
return FloatValue(1.0)
elif len(sh) == 1:
# Vector v makes dv/dv the identity matrix
return Identity(sh[0])
else:
# TODO: Add a type for this higher order identity?
# II[i0,i1,i2,j0,j1,j2] = 1 if all((i0==j0, i1==j1, i2==j2)) else 0
# Tensor v makes dv/dv some kind of higher rank identity tensor
ind1 = ()
ind2 = ()
for d in sh:
i, j = indices(2)
dij = Identity(d)[i, j]
if res is None:
res = dij
else:
res *= dij
ind1 += (i,)
ind2 += (j,)
fp = as_tensor(res, ind1 + ind2)
return fp
def indexed(self, o, Ap, ii): # TODO: (Partially) duplicated in nesting rules
# Propagate zeros
if isinstance(Ap, Zero):
return self.independent_operator(o)
# Untangle as_tensor(C[kk], jj)[ii] -> C[ll] to simplify
# resulting expression
if isinstance(Ap, ComponentTensor):
B, jj = Ap.ufl_operands
if isinstance(B, Indexed):
C, kk = B.ufl_operands
kk = list(kk)
if all(j in kk for j in jj):
Cind = list(kk)
for i, j in zip(ii, jj):
Cind[kk.index(j)] = i
return Indexed(C, MultiIndex(tuple(Cind)))
# Otherwise a more generic approach
r = len(Ap.ufl_shape) - len(ii)
if r:
kk = indices(r)
op = Indexed(Ap, MultiIndex(ii.indices() + kk))
op = as_tensor(op, kk)
else:
op = Indexed(Ap, ii)
return op
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 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 inner(self, o, a, b):
ash = a.ufl_shape
bsh = b.ufl_shape
if ash != bsh:
error("Nonmatching shapes.")
ii = indices(len(ash))
# Creates multiple IndexSums over a Product
s = a[ii] * Conj(b[ii])
return s
def inner(self, o, a, b):
ash = a.ufl_shape
bsh = b.ufl_shape
if ash != bsh:
error("Nonmatching shapes.")
ii = indices(len(ash))
# Creates multiple IndexSums over a Product
s = a[ii]*Conj(b[ii])
return s
def facet_jacobian(self, o):
if self._preserve_types[o._ufl_typecode_]:
return o
domain = o.ufl_domain()
J = self.jacobian(Jacobian(domain))
RFJ = CellFacetJacobian(domain)
i, j, k = indices(3)
return as_tensor(J[i, k] * RFJ[k, j], (i, j))
def _div(self, o):
if not isinstance(o, _valid_types):
return NotImplemented
sh = self.ufl_shape
if sh:
ii = indices(len(sh))
d = Division(self[ii], o)
return as_tensor(d, ii)
return Division(self, o)
def facet_jacobian(self, o):
if self._preserve_types[o._ufl_typecode_]:
return o
domain = o.ufl_domain()
J = self.jacobian(Jacobian(domain))
RFJ = CellFacetJacobian(domain)
i, j, k = indices(3)
return as_tensor(J[i, k]*RFJ[k, j], (i, j))
def grad_to_reference_grad(o, K):
"""Relates grad(o) to reference_grad(o) using the Jacobian inverse.
Args
----
o: Operand
K: Jacobian inverse
Returns
-------
Do: grad(o) written in terms of reference_grad(o) and K
"""
r = indices(len(o.ufl_shape))
i, j = indices(2)
# grad(o) == K_ji rgrad(o)_rj
Do = as_tensor(K[j, i] * ReferenceGrad(o)[r + (j,)], r + (i,))
return Do
def _div(self, o):
if not isinstance(o, _valid_types):
return NotImplemented
sh = self.ufl_shape
if sh:
ii = indices(len(sh))
d = Division(self[ii], o)
return as_tensor(d, ii)
return Division(self, o)
def pseudo_inverse_expr(A):
"""Compute the Penrose-Moore pseudo-inverse of A: (A.T*A)^-1 * A.T."""
m, n = A.ufl_shape
if n == 1:
# Simpler special case for 1d
i, j, k = indices(3)
return as_tensor(A[i, j], (j, i)) / (A[k, 0] * A[k, 0])
else:
# Generic formulation
return generic_pseudo_inverse_expr(A)
def pseudo_inverse_expr(A):
"""Compute the Penrose-Moore pseudo-inverse of A: (A.T*A)^-1 * A.T."""
m, n = A.ufl_shape
if n == 1:
# Simpler special case for 1d
i, j, k = indices(3)
return as_tensor(A[i, j], (j, i)) / (A[k, 0] * A[k, 0])
else:
# Generic formulation
return generic_pseudo_inverse_expr(A)
def as_scalars(*expressions):
"""Given multiple scalar or tensor valued expressions A, returns either of the tuples::
(a,b) = (A, ())
(a,b) = ([A[0][indices], ..., A[-1][indices]], indices)
such that a is always a list of scalar valued expressions."""
ii = indices(len(expressions[0].ufl_shape))
if ii:
expressions = [expression[ii] for expression in expressions]
return expressions, ii
def as_scalar(expression):
"""Given a scalar or tensor valued expression A, returns either of the tuples::
(a,b) = (A, ())
(a,b) = (A[indices], indices)
such that a is always a scalar valued expression."""
ii = indices(len(expression.ufl_shape))
if ii:
expression = expression[ii]
return expression, ii
def as_scalars(*expressions):
"""Given multiple scalar or tensor valued expressions A, returns either of the tuples::
(a,b) = (A, ())
(a,b) = ([A[0][indices], ..., A[-1][indices]], indices)
such that a is always a list of scalar valued expressions."""
ii = indices(len(expressions[0].ufl_shape))
if ii:
expressions = [expression[ii] for expression in expressions]
return expressions, ii
def as_scalar(expression):
"""Given a scalar or tensor valued expression A, returns either of the tuples::
(a,b) = (A, ())
(a,b) = (A[indices], indices)
such that a is always a scalar valued expression."""
ii = indices(len(expression.ufl_shape))
if ii:
expression = expression[ii]
return expression, ii
def cell_coordinate(self, o):
"Compute from physical coordinates if they are known, using the appropriate mappings."
if self._preserve_types[o._ufl_typecode_]:
return o
domain = o.ufl_domain()
K = self.jacobian_inverse(JacobianInverse(domain))
x = self.spatial_coordinate(SpatialCoordinate(domain))
x0 = CellOrigin(domain)
i, j = indices(2)
X = as_tensor(K[i, j] * (x[j] - x0[j]), (i,))
return X
def cell_coordinate(self, o):
"Compute from physical coordinates if they are known, using the appropriate mappings."
if self._preserve_types[o._ufl_typecode_]:
return o
domain = o.ufl_domain()
K = self.jacobian_inverse(JacobianInverse(domain))
x = self.spatial_coordinate(SpatialCoordinate(domain))
x0 = CellOrigin(domain)
i, j = indices(2)
X = as_tensor(K[i, j] * (x[j] - x0[j]), (i, ))
return X
def contraction(a, a_axes, b, b_axes):
"UFL operator: Take the contraction of a and b over given axes."
ai, bi = a_axes, b_axes
if len(ai) != len(bi):
error("Contraction must be over the same number of axes.")
ash = a.ufl_shape
bsh = b.ufl_shape
aii = indices(len(a.ufl_shape))
bii = indices(len(b.ufl_shape))
cii = indices(len(ai))
shape = [None] * len(ai)
for i, j in enumerate(ai):
aii[j] = cii[i]
shape[i] = ash[j]
for i, j in enumerate(bi):
bii[j] = cii[i]
if shape[i] != bsh[j]:
error("Shape mismatch in contraction.")
s = a[aii] * b[bii]
cii = set(cii)
ii = tuple(i for i in (aii + bii) if i not in cii)
return as_tensor(s, ii)
def compute_gprimeterm(ngrads, vval, vcomp, wshape, wcomp):
# Apply gradients directly to argument vval, and get the
# right indexed scalar component(s)
kk = indices(ngrads)
Dvkk = apply_grads(vval)[vcomp+kk]
# Place scalar component(s) Dvkk into the right tensor
# positions
if wshape:
Ejj, jj = unit_indexed_tensor(wshape, wcomp)
else:
Ejj, jj = 1, ()
gprimeterm = as_tensor(Ejj*Dvkk, jj+kk)
return gprimeterm
def contraction(a, a_axes, b, b_axes):
"UFL operator: Take the contraction of a and b over given axes."
ai, bi = a_axes, b_axes
if len(ai) != len(bi):
error("Contraction must be over the same number of axes.")
ash = a.ufl_shape
bsh = b.ufl_shape
aii = indices(len(a.ufl_shape))
bii = indices(len(b.ufl_shape))
cii = indices(len(ai))
shape = [None] * len(ai)
for i, j in enumerate(ai):
aii[j] = cii[i]
shape[i] = ash[j]
for i, j in enumerate(bi):
bii[j] = cii[i]
if shape[i] != bsh[j]:
error("Shape mismatch in contraction.")
s = a[aii] * b[bii]
cii = set(cii)
ii = tuple(i for i in (aii + bii) if i not in cii)
return as_tensor(s, ii)
def compute_gprimeterm(ngrads, vval, vcomp, wshape, wcomp):
# Apply gradients directly to argument vval, and get the
# right indexed scalar component(s)
kk = indices(ngrads)
Dvkk = apply_grads(vval)[vcomp + kk]
# Place scalar component(s) Dvkk into the right tensor
# positions
if wshape:
Ejj, jj = unit_indexed_tensor(wshape, wcomp)
else:
Ejj, jj = 1, ()
gprimeterm = as_tensor(Ejj * Dvkk, jj + kk)
return gprimeterm
def unit_indexed_tensor(shape, component):
from ufl.constantvalue import Identity
from ufl.operators import outer # a bit of circular dependency issue here
r = len(shape)
if r == 0:
return 0, ()
jj = indices(r)
es = []
for i in range(r):
s = shape[i]
c = component[i]
j = jj[i]
e = Identity(s)[c, j]
es.append(e)
E = es[0]
for e in es[1:]:
E = outer(E, e)
return E, jj
def unit_indexed_tensor(shape, component):
from ufl.constantvalue import Identity
from ufl.operators import outer # a bit of circular dependency issue here
r = len(shape)
if r == 0:
return 0, ()
jj = indices(r)
es = []
for i in range(r):
s = shape[i]
c = component[i]
j = jj[i]
e = Identity(s)[c, j]
es.append(e)
E = es[0]
for e in es[1:]:
E = outer(E, e)
return E, jj
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 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 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 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 _mapped(self, t):
# Check that we have a valid input object
if not isinstance(t, Terminal):
error("Expecting a Terminal.")
# Get modifiers accumulated by previous handler calls
ngrads = self._ngrads
restricted = self._restricted
avg = self._avg
if avg != "":
error("Averaging not implemented.") # FIXME
# These are the global (g) and reference (r) values
if isinstance(t, FormArgument):
g = t
r = ReferenceValue(g)
elif isinstance(t, GeometricQuantity):
g = t
r = g
else:
error("Unexpected type {0}.".format(type(t).__name__))
# Some geometry mapping objects we may need multiple times below
domain = t.ufl_domain()
J = Jacobian(domain)
detJ = JacobianDeterminant(domain)
K = JacobianInverse(domain)
# Restrict geometry objects if applicable
if restricted:
J = J(restricted)
detJ = detJ(restricted)
K = K(restricted)
# Create Hdiv mapping from possibly restricted geometry objects
Mdiv = (1.0 / detJ) * J
# Get component indices of global and reference terminal objects
gtsh = g.ufl_shape
# rtsh = r.ufl_shape
gtcomponents = compute_indices(gtsh)
# rtcomponents = compute_indices(rtsh)
# Create core modified terminal, with eventual
# layers of grad applied directly to the terminal,
# then eventual restriction applied last
for i in range(ngrads):
g = Grad(g)
r = ReferenceGrad(r)
if restricted:
g = g(restricted)
r = r(restricted)
# Get component indices of global and reference objects with
# grads applied
gsh = g.ufl_shape
# rsh = r.ufl_shape
# gcomponents = compute_indices(gsh)
# rcomponents = compute_indices(rsh)
# Get derivative component indices
dsh = gsh[len(gtsh):]
dcomponents = compute_indices(dsh)
# Create nested array to hold expressions for global
# components mapped from reference values
def ndarray(shape):
if len(shape) == 0:
return [None]
elif len(shape) == 1:
return [None] * shape[-1]
else:
return [ndarray(shape[1:]) for i in range(shape[0])]
global_components = ndarray(gsh)
# Compute mapping from reference values for each global component
for gtc in gtcomponents:
if isinstance(t, FormArgument):
# Find basic subelement and element-local component
# ec, element, eoffset = t.ufl_element().extract_component2(gtc) # FIXME: Translate this correctly
eoffset = 0
ec, element = t.ufl_element().extract_reference_component(gtc)
# Select mapping M from element, pick row emapping =
# M[ec,:], or emapping = [] if no mapping
if isinstance(element, MixedElement):
error("Expecting a basic element here.")
mapping = element.mapping()
if mapping == "contravariant Piola": # S == HDiv:
# Handle HDiv elements with contravariant piola
# mapping contravariant_hdiv_mapping = (1/det J) *
# J * PullbackOf(o)
ec, = ec
emapping = Mdiv[ec, :]
elif mapping == "covariant Piola": # S == HCurl:
# Handle HCurl elements with covariant piola mapping
# covariant_hcurl_mapping = JinvT * PullbackOf(o)
ec, = ec
emapping = K[:, ec] # Column of K is row of K.T
elif mapping == "identity":
emapping = None
else:
error("Unknown mapping {0}".format(mapping))
elif isinstance(t, GeometricQuantity):
eoffset = 0
emapping = None
else:
error("Unexpected type {0}.".format(type(t).__name__))
# Create indices
# if rtsh:
# i = Index()
if len(dsh) != ngrads:
error("Mismatch between derivative shape and ngrads.")
if ngrads:
ii = indices(ngrads)
else:
ii = ()
# Apply mapping row to reference object
if emapping: # Mapped, always nonscalar terminal Not
# using IndexSum for the mapping row dot product to
# keep it simple, because we don't have a slice type
emapped_ops = [emapping[s] * Indexed(r, MultiIndex((FixedIndex(eoffset + s),) + ii))
for s in range(len(emapping))]
emapped = sum(emapped_ops[1:], emapped_ops[0])
elif gtc: # Nonscalar terminal, unmapped
emapped = Indexed(r, MultiIndex((FixedIndex(eoffset),) + ii))
elif ngrads: # Scalar terminal, unmapped, with derivatives
emapped = Indexed(r, MultiIndex(ii))
else: # Scalar terminal, unmapped, no derivatives
emapped = r
for di in dcomponents:
# Multiply derivative mapping rows, parameterized by
# free column indices
dmapping = as_ufl(1)
for j in range(ngrads):
dmapping *= K[ii[j], di[j]] # Row of K is column of JinvT
# Compute mapping from reference values for this
# particular global component
global_value = dmapping * emapped
# Apply index sums
# if rtsh:
# global_value = IndexSum(global_value, MultiIndex((i,)))
# for j in range(ngrads): # Applied implicitly in the dmapping * emapped above
# global_value = IndexSum(global_value, MultiIndex((ii[j],)))
# This is the component index into the full object
# with grads applied
gc = gtc + di
# Insert in nested list
comp = global_components
for i in gc[:-1]:
comp = comp[i]
comp[0 if gc == () else gc[-1]] = global_value
# Wrap nested list in as_tensor unless we have a scalar
# expression
if gsh:
tensor = as_tensor(global_components)
else:
tensor, = global_components
return tensor
def grad(self, o):
# Peel off the Grads and count them, and get restriction if
# it's between the grad and the terminal
ngrads = 0
restricted = ''
rv = False
while not o._ufl_is_terminal_:
if isinstance(o, Grad):
o, = o.ufl_operands
ngrads += 1
elif isinstance(o, Restricted):
restricted = o.side()
o, = o.ufl_operands
elif isinstance(o, ReferenceValue):
rv = True
o, = o.ufl_operands
else:
error("Invalid type %s" % o._ufl_class_.__name__)
f = o
if rv:
f = ReferenceValue(f)
# Get domain and create Jacobian inverse object
domain = o.ufl_domain()
Jinv = JacobianInverse(domain)
if is_cellwise_constant(Jinv):
# Optimise slightly by turning Grad(Grad(...)) into
# J^(-T)J^(-T)RefGrad(RefGrad(...))
# rather than J^(-T)RefGrad(J^(-T)RefGrad(...))
# Create some new indices
ii = indices(len(f.ufl_shape)) # Indices to get to the scalar component of f
jj = indices(ngrads) # Indices to sum over the local gradient axes with the inverse Jacobian
kk = indices(ngrads) # Indices for the leftover inverse Jacobian axes
# Preserve restricted property
if restricted:
Jinv = Jinv(restricted)
f = f(restricted)
# Apply the same number of ReferenceGrad without mappings
lgrad = f
for i in range(ngrads):
lgrad = ReferenceGrad(lgrad)
# Apply mappings with scalar indexing operations (assumes
# ReferenceGrad(Jinv) is zero)
jinv_lgrad_f = lgrad[ii + jj]
for j, k in zip(jj, kk):
jinv_lgrad_f = Jinv[j, k] * jinv_lgrad_f
# Wrap back in tensor shape, derivative axes at the end
jinv_lgrad_f = as_tensor(jinv_lgrad_f, ii + kk)
else:
# J^(-T)RefGrad(J^(-T)RefGrad(...))
# Preserve restricted property
if restricted:
Jinv = Jinv(restricted)
f = f(restricted)
jinv_lgrad_f = f
for foo in range(ngrads):
ii = indices(len(jinv_lgrad_f.ufl_shape)) # Indices to get to the scalar component of f
j, k = indices(2)
lgrad = ReferenceGrad(jinv_lgrad_f)
jinv_lgrad_f = Jinv[j, k] * lgrad[ii + (j,)]
# Wrap back in tensor shape, derivative axes at the end
jinv_lgrad_f = as_tensor(jinv_lgrad_f, ii + (k,))
return jinv_lgrad_f
def sym(self, o, A):
i, j = indices(2)
return as_matrix((A[i, j] + A[j, i]) / 2, (i, j))
def transposed(self, o, A):
i, j = indices(2)
return as_tensor(A[i, j], (j, i))
def generic_pseudo_determinant_expr(A):
"""Compute the pseudo-determinant of A: sqrt(det(A.T*A))."""
i, j, k = indices(3)
ATA = as_tensor(A[k, i] * A[k, j], (i, j))
return sqrt(determinant_expr(ATA))
def outer(self, o, a, b):
ii = indices(len(a.ufl_shape))
jj = indices(len(b.ufl_shape))
# Create a Product with no shared indices
s = Conj(a[ii]) * b[jj]
return as_tensor(s, ii + jj)
def outer(self, o, a, b):
ii = indices(len(a.ufl_shape))
jj = indices(len(b.ufl_shape))
# Create a Product with no shared indices
s = Conj(a[ii])*b[jj]
return as_tensor(s, ii+jj)
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 transposed(self, o, A):
i, j = indices(2)
return as_tensor(A[i, j], (j, 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
def sym(self, o, A):
i, j = indices(2)
return as_matrix((A[i, j] + A[j, i]) / 2, (i, j))