# Source code for ufl.tensoralgebra

```"""Compound tensor algebra operations."""
# Copyright (C) 2008-2016 Martin Sandve Alnæs
#
# This file is part of UFL (https://www.fenicsproject.org)
#

from ufl.algebra import Conj, Operator
from ufl.constantvalue import Zero
from ufl.core.expr import ufl_err_str
from ufl.core.ufl_type import ufl_type
from ufl.index_combination_utils import merge_nonoverlapping_indices
from ufl.precedence import parstr
from ufl.sorting import sorted_expr

# Algebraic operations on tensors:
# FloatValues:
#   dot(a,b)      = a*b
#   inner(a,b)    = a*b
#   outer(a,b)    = a*b
# Vectors:
#   dot(u,v)      = u_i v_i
#   inner(u,v)    = u_i v_i
#   outer(u,v)    = A | A_ij = u_i v_j
# Matrices:
#   dot(A,B)      = C | C_ij = A_{ik} B_{kj}
#   inner(A,B)    = A_{ij} B_{ij}
#   outer(A,B)    = C | C_ijkl = A_ij B_kl
# Combined:
#   dot(A,u)      = v | v_i = A_{ik} u_k
#   inner(A,u)    = v | v_i = A_{ik} u_k
#   outer(A,u)    = C | C_ijk = B_ij u_k
#   dot(u,B)      = v | v_i = u_k B_{ki}
#   inner(u,B)    = v | v_i = u_k B_{ki}
#   outer(u,B)    = C | C_ijk = u_i B_jk
#
# Argument requirements:
#   dot(x,y):   last index of x has same dimension as first index of y
#   inner(x,y): shape of x equals the shape of y

# --- Classes representing compound tensor algebra operations ---

[docs]@ufl_type(is_abstract=True)
class CompoundTensorOperator(Operator):
"""Compount tensor operator."""

__slots__ = ()

def __init__(self, operands):
"""Initialise."""
Operator.__init__(self, operands)

# TODO: Use this and make Sum handle scalars only?
#       This would simplify some algorithms. The only
#       problem is we can't use + in many algorithms because
#       this type should be expanded by expand_compounds.
# class TensorSum(CompoundTensorOperator):
#     "Sum of nonscalar expressions."
#     pass

# TODO: Use this similarly to TensorSum?
#       This would simplify some algorithms. The only
#       problem is we can't use / in many algorithms because
#       this type should be expanded by expand_compounds.
# class TensorDivision(CompoundTensorOperator):
#     "Division of nonscalar expression with a scalar expression."
#     pass

# TODO: Use this similarly to TensorSum?
#       This would simplify some algorithms. The only
#       problem is we can't use * in many algorithms because
#       this type should be expanded by expand_compounds.
# class MatrixProduct(CompoundTensorOperator):
#     "Product of a matrix with a matrix or vector."
#     pass

# TODO: Use this similarly to TensorSum?
#       This would simplify some algorithms. The only
#       problem is we can't use abs in many algorithms because
#       this type should be expanded by expand_compounds.
# class TensorAbs(CompoundTensorOperator):
#     "Absolute value of nonscalar expression."
#     pass

[docs]@ufl_type(is_shaping=True, num_ops=1, inherit_indices_from_operand=0)
class Transposed(CompoundTensorOperator):
"""Transposed tensor."""

__slots__ = ()

def __new__(cls, A):
"""Create new Transposed."""
if isinstance(A, Zero):
a, b = A.ufl_shape
return Zero((b, a), A.ufl_free_indices, A.ufl_index_dimensions)
return CompoundTensorOperator.__new__(cls)

def __init__(self, A):
"""Initialise."""
CompoundTensorOperator.__init__(self, (A,))
if len(A.ufl_shape) != 2:
raise ValueError("Transposed is only defined for rank 2 tensors.")

@property
def ufl_shape(self):
"""Get the UFL shape."""
s = self.ufl_operands[0].ufl_shape
return (s[1], s[0])

def __str__(self):
"""Format as a string."""
return "%s^T" % parstr(self.ufl_operands[0], self)

[docs]@ufl_type(num_ops=2)
class Outer(CompoundTensorOperator):
"""Outer."""

__slots__ = ("ufl_free_indices", "ufl_index_dimensions")

def __new__(cls, a, b):
"""Create new Outer."""
ash, bsh = a.ufl_shape, b.ufl_shape
if isinstance(a, Zero) or isinstance(b, Zero):
fi, fid = merge_nonoverlapping_indices(a, b)
return Zero(ash + bsh, fi, fid)
if ash == () or bsh == ():
return Conj(a) * b
return CompoundTensorOperator.__new__(cls)

def __init__(self, a, b):
"""Initialise."""
CompoundTensorOperator.__init__(self, (a, b))
fi, fid = merge_nonoverlapping_indices(a, b)
self.ufl_free_indices = fi
self.ufl_index_dimensions = fid

@property
def ufl_shape(self):
"""Get the UFL shape."""
return self.ufl_operands[0].ufl_shape + self.ufl_operands[1].ufl_shape

def __str__(self):
"""Format as a string."""
return "%s (X) %s" % (
parstr(self.ufl_operands[0], self),
parstr(self.ufl_operands[1], self),
)

[docs]@ufl_type(num_ops=2)
class Inner(CompoundTensorOperator):
"""Inner."""

__slots__ = ("ufl_free_indices", "ufl_index_dimensions")

def __new__(cls, a, b):
"""Create new Inner."""
# Checks
ash, bsh = a.ufl_shape, b.ufl_shape
if ash != bsh:
raise ValueError(f"Shapes do not match: {ufl_err_str(a)} and {ufl_err_str(b)}")

# Simplification
if isinstance(a, Zero) or isinstance(b, Zero):
fi, fid = merge_nonoverlapping_indices(a, b)
return Zero((), fi, fid)
elif ash == ():
return a * Conj(b)
# sort operands for unique representation,
# must be independent of various counts etc.
# as explained in cmp_expr
if (a, b) != tuple(sorted_expr((a, b))):
return Conj(Inner(b, a))

return CompoundTensorOperator.__new__(cls)

def __init__(self, a, b):
"""Initialise."""
CompoundTensorOperator.__init__(self, (a, b))

fi, fid = merge_nonoverlapping_indices(a, b)
self.ufl_free_indices = fi
self.ufl_index_dimensions = fid

ufl_shape = ()

def __str__(self):
"""Format as a string."""
return "%s : %s" % (parstr(self.ufl_operands[0], self), parstr(self.ufl_operands[1], self))

[docs]@ufl_type(num_ops=2)
class Dot(CompoundTensorOperator):
"""Dot."""

__slots__ = ("ufl_free_indices", "ufl_index_dimensions")

def __new__(cls, a, b):
"""Create new Dot."""
ash = a.ufl_shape
bsh = b.ufl_shape
ar, br = len(ash), len(bsh)
scalar = ar == 0 and br == 0

# Checks
if not ((ar >= 1 and br >= 1) or scalar):
raise ValueError(
"Dot product requires non-scalar arguments, "
f"got arguments with ranks {ar} and {br}."
)
if not (scalar or ash[-1] == bsh[0]):
raise ValueError("Dimension mismatch in dot product.")

# Simplification
if isinstance(a, Zero) or isinstance(b, Zero):
shape = ash[:-1] + bsh[1:]
fi, fid = merge_nonoverlapping_indices(a, b)
return Zero(shape, fi, fid)
elif scalar:  # TODO: Move this to def dot()?
return a * b

return CompoundTensorOperator.__new__(cls)

def __init__(self, a, b):
"""Initialise."""
CompoundTensorOperator.__init__(self, (a, b))
fi, fid = merge_nonoverlapping_indices(a, b)
self.ufl_free_indices = fi
self.ufl_index_dimensions = fid

@property
def ufl_shape(self):
"""Get the UFL shape."""
return self.ufl_operands[0].ufl_shape[:-1] + self.ufl_operands[1].ufl_shape[1:]

def __str__(self):
"""Format as a string."""
return "%s . %s" % (parstr(self.ufl_operands[0], self), parstr(self.ufl_operands[1], self))

[docs]@ufl_type(is_index_free=True, num_ops=1)
class Perp(CompoundTensorOperator):
"""Perp."""

__slots__ = ()

def __new__(cls, A):
"""Create new Perp."""
sh = A.ufl_shape

# Checks
if not len(sh) == 1:
raise ValueError(f"Perp requires arguments of rank 1, got {ufl_err_str(A)}")
if not sh[0] == 2:
raise ValueError(f"Perp can only work on 2D vectors, got {ufl_err_str(A)}")

# Simplification
if isinstance(A, Zero):
return Zero(sh, A.ufl_free_indices, A.ufl_index_dimensions)

return CompoundTensorOperator.__new__(cls)

def __init__(self, A):
"""Initialise."""
CompoundTensorOperator.__init__(self, (A,))

ufl_shape = (2,)

def __str__(self):
"""Format as a string."""
return "perp(%s)" % self.ufl_operands[0]

[docs]@ufl_type(num_ops=2)
class Cross(CompoundTensorOperator):
"""Cross."""

__slots__ = ("ufl_free_indices", "ufl_index_dimensions")

def __new__(cls, a, b):
"""Create new Cross."""
ash = a.ufl_shape
bsh = b.ufl_shape

# Checks
if not (len(ash) == 1 and ash == bsh):
raise ValueError(
f"Cross product requires arguments of rank 1, got {ufl_err_str(a)} "
f"and {ufl_err_str(b)}."
)

# Simplification
if isinstance(a, Zero) or isinstance(b, Zero):
fi, fid = merge_nonoverlapping_indices(a, b)
return Zero(ash, fi, fid)

return CompoundTensorOperator.__new__(cls)

def __init__(self, a, b):
"""Initialise."""
CompoundTensorOperator.__init__(self, (a, b))
fi, fid = merge_nonoverlapping_indices(a, b)
self.ufl_free_indices = fi
self.ufl_index_dimensions = fid

ufl_shape = (3,)

def __str__(self):
"""Format as a string."""
return "%s x %s" % (parstr(self.ufl_operands[0], self), parstr(self.ufl_operands[1], self))

[docs]@ufl_type(num_ops=1, inherit_indices_from_operand=0)
class Trace(CompoundTensorOperator):
"""Trace."""

__slots__ = ()

def __new__(cls, A):
"""Create new Trace."""
# Checks
if len(A.ufl_shape) != 2:
raise ValueError("Trace of tensor with rank != 2 is undefined.")

# Simplification
if isinstance(A, Zero):
return Zero((), A.ufl_free_indices, A.ufl_index_dimensions)

return CompoundTensorOperator.__new__(cls)

def __init__(self, A):
"""Initialise."""
CompoundTensorOperator.__init__(self, (A,))

ufl_shape = ()

def __str__(self):
"""Format as a string."""
return "tr(%s)" % self.ufl_operands[0]

[docs]@ufl_type(is_scalar=True, num_ops=1)
class Determinant(CompoundTensorOperator):
"""Determinant."""

__slots__ = ()

def __new__(cls, A):
"""Create new Determinant."""
sh = A.ufl_shape
r = len(sh)
Afi = A.ufl_free_indices

# Checks
if r not in (0, 2):
raise ValueError("Determinant of tensor with rank != 2 is undefined.")
if r == 2 and sh[0] != sh[1]:
raise ValueError("Cannot take determinant of rectangular rank 2 tensor.")
if Afi:
raise ValueError("Not expecting free indices in determinant.")

# Simplification
if isinstance(A, Zero):
return Zero((), Afi, A.ufl_index_dimensions)
if r == 0:
return A

return CompoundTensorOperator.__new__(cls)

def __init__(self, A):
"""Initialise."""
CompoundTensorOperator.__init__(self, (A,))

def __str__(self):
"""Format as a string."""
return "det(%s)" % self.ufl_operands[0]

# TODO: Drop Inverse and represent it as product of Determinant and
# Cofactor?
[docs]@ufl_type(is_index_free=True, num_ops=1)
class Inverse(CompoundTensorOperator):
"""Inverse."""

__slots__ = ()

def __new__(cls, A):
"""Create new Inverse."""
sh = A.ufl_shape
r = len(sh)

# Checks
if A.ufl_free_indices:
raise ValueError("Not expecting free indices in Inverse.")
if isinstance(A, Zero):
raise ValueError("Division by zero!")

# Simplification
if r == 0:
return 1 / A

# More checks
if r != 2:
raise ValueError("Inverse of tensor with rank != 2 is undefined.")
if sh[0] != sh[1]:
raise ValueError(f"Cannot take inverse of rectangular matrix with dimensions {sh}.")

return CompoundTensorOperator.__new__(cls)

def __init__(self, A):
"""Initialise."""
CompoundTensorOperator.__init__(self, (A,))

@property
def ufl_shape(self):
"""Get the UFL shape."""
return self.ufl_operands[0].ufl_shape

def __str__(self):
"""Format as a string."""
return "%s^-1" % parstr(self.ufl_operands[0], self)

[docs]@ufl_type(is_index_free=True, num_ops=1)
class Cofactor(CompoundTensorOperator):
"""Cofactor."""

__slots__ = ()

def __init__(self, A):
"""Initialise."""
CompoundTensorOperator.__init__(self, (A,))

# Checks
sh = A.ufl_shape
if len(sh) != 2:
raise ValueError("Cofactor of tensor with rank != 2 is undefined.")
if sh[0] != sh[1]:
raise ValueError(f"Cannot take cofactor of rectangular matrix with dimensions {sh}.")
if A.ufl_free_indices:
raise ValueError("Not expecting free indices in Cofactor.")
if isinstance(A, Zero):
raise ValueError("Cannot take cofactor of zero matrix.")

@property
def ufl_shape(self):
"""Get the UFL shape."""
return self.ufl_operands[0].ufl_shape

def __str__(self):
"""Format as a string."""
return "cofactor(%s)" % self.ufl_operands[0]

[docs]@ufl_type(num_ops=1, inherit_shape_from_operand=0, inherit_indices_from_operand=0)
class Deviatoric(CompoundTensorOperator):
"""Deviatoric."""

__slots__ = ()

def __new__(cls, A):
"""Create new Deviatoric."""
sh = A.ufl_shape

# Checks
if len(sh) != 2:
raise ValueError("Deviatoric part of tensor with rank != 2 is undefined.")
if sh[0] != sh[1]:
raise ValueError(
f"Cannot take deviatoric part of rectangular matrix with dimensions {sh}."
)
if A.ufl_free_indices:
raise ValueError("Not expecting free indices in Deviatoric.")

# Simplification
if isinstance(A, Zero):
return Zero(A.ufl_shape, A.ufl_free_indices, A.ufl_index_dimensions)

return CompoundTensorOperator.__new__(cls)

def __init__(self, A):
"""Initialise."""
CompoundTensorOperator.__init__(self, (A,))

def __str__(self):
"""Format as a string."""
return "dev(%s)" % self.ufl_operands[0]

[docs]@ufl_type(num_ops=1, inherit_shape_from_operand=0, inherit_indices_from_operand=0)
class Skew(CompoundTensorOperator):
"""Skew."""

__slots__ = ()

def __new__(cls, A):
"""Create new Skew."""
sh = A.ufl_shape
Afi = A.ufl_free_indices

# Checks
if len(sh) != 2:
raise ValueError("Skew symmetric part of tensor with rank != 2 is undefined.")
if sh[0] != sh[1]:
raise ValueError(f"Cannot take skew part of rectangular matrix with dimensions {sh}.")
if Afi:
raise ValueError("Not expecting free indices in Skew.")

# Simplification
if isinstance(A, Zero):
return Zero(A.ufl_shape, Afi, A.ufl_index_dimensions)

return CompoundTensorOperator.__new__(cls)

def __init__(self, A):
"""Initialise."""
CompoundTensorOperator.__init__(self, (A,))

def __str__(self):
"""Format as a string."""
return "skew(%s)" % self.ufl_operands[0]

[docs]@ufl_type(num_ops=1, inherit_shape_from_operand=0, inherit_indices_from_operand=0)
class Sym(CompoundTensorOperator):
"""Sym."""

__slots__ = ()

def __new__(cls, A):
"""Create new Sym."""
sh = A.ufl_shape
Afi = A.ufl_free_indices

# Checks
if len(sh) != 2:
raise ValueError("Symmetric part of tensor with rank != 2 is undefined.")
if sh[0] != sh[1]:
raise ValueError(
f"Cannot take symmetric part of rectangular matrix with dimensions {sh}."
)
if Afi:
raise ValueError("Not expecting free indices in Sym.")

# Simplification
if isinstance(A, Zero):
return Zero(A.ufl_shape, Afi, A.ufl_index_dimensions)

return CompoundTensorOperator.__new__(cls)

def __init__(self, A):
"""Initialise."""
CompoundTensorOperator.__init__(self, (A,))

def __str__(self):
"""Format as a string."""
return f"sym({self.ufl_operands[0]})"
```