Source code for ufl.tensoralgebra

# -*- coding: utf-8 -*-
"""Compound tensor algebra operations."""

# Copyright (C) 2008-2016 Martin Sandve Aln├Žs
#
# This file is part of UFL (https://www.fenicsproject.org)
#
# SPDX-License-Identifier:    LGPL-3.0-or-later

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

# 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): __slots__ = () def __init__(self, operands): 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): __slots__ = () def __new__(cls, A): 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): CompoundTensorOperator.__init__(self, (A,)) if len(A.ufl_shape) != 2: error("Transposed is only defined for rank 2 tensors.") @property def ufl_shape(self): s = self.ufl_operands[0].ufl_shape return (s[1], s[0]) def __str__(self): return "%s^T" % parstr(self.ufl_operands[0], self)
[docs]@ufl_type(num_ops=2) class Outer(CompoundTensorOperator): __slots__ = ("ufl_free_indices", "ufl_index_dimensions") def __new__(cls, a, b): 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): 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): return self.ufl_operands[0].ufl_shape + self.ufl_operands[1].ufl_shape def __str__(self): 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): __slots__ = ("ufl_free_indices", "ufl_index_dimensions") def __new__(cls, a, b): # Checks ash, bsh = a.ufl_shape, b.ufl_shape if ash != bsh: error("Shapes do not match: %s and %s." % (ufl_err_str(a), 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): 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): return "%s : %s" % (parstr(self.ufl_operands[0], self), parstr(self.ufl_operands[1], self))
[docs]@ufl_type(num_ops=2) class Dot(CompoundTensorOperator): __slots__ = ("ufl_free_indices", "ufl_index_dimensions") def __new__(cls, a, b): 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): error("Dot product requires non-scalar arguments, " "got arguments with ranks %d and %d." % (ar, br)) if not (scalar or ash[-1] == bsh[0]): error("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): 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): return self.ufl_operands[0].ufl_shape[:-1] + self.ufl_operands[1].ufl_shape[1:] def __str__(self): return "%s . %s" % (parstr(self.ufl_operands[0], self), parstr(self.ufl_operands[1], self))
[docs]@ufl_type(num_ops=2) class Cross(CompoundTensorOperator): __slots__ = ("ufl_free_indices", "ufl_index_dimensions") def __new__(cls, a, b): ash = a.ufl_shape bsh = b.ufl_shape # Checks if not (len(ash) == 1 and ash == bsh): error("Cross product requires arguments of rank 1, got %s and %s." % ( ufl_err_str(a), 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): 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): 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): __slots__ = () def __new__(cls, A): # Checks if len(A.ufl_shape) != 2: error("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): CompoundTensorOperator.__init__(self, (A,)) ufl_shape = () def __str__(self): return "tr(%s)" % self.ufl_operands[0]
[docs]@ufl_type(is_scalar=True, num_ops=1) class Determinant(CompoundTensorOperator): __slots__ = () def __new__(cls, A): sh = A.ufl_shape r = len(sh) Afi = A.ufl_free_indices # Checks if r not in (0, 2): error("Determinant of tensor with rank != 2 is undefined.") if r == 2 and sh[0] != sh[1]: error("Cannot take determinant of rectangular rank 2 tensor.") if Afi: error("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): CompoundTensorOperator.__init__(self, (A,)) def __str__(self): 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): __slots__ = () def __new__(cls, A): sh = A.ufl_shape r = len(sh) # Checks if A.ufl_free_indices: error("Not expecting free indices in Inverse.") if isinstance(A, Zero): error("Division by zero!") # Simplification if r == 0: return 1 / A # More checks if r != 2: error("Inverse of tensor with rank != 2 is undefined.") if sh[0] != sh[1]: error("Cannot take inverse of rectangular matrix with dimensions %s." % (sh,)) return CompoundTensorOperator.__new__(cls) def __init__(self, A): CompoundTensorOperator.__init__(self, (A,)) @property def ufl_shape(self): return self.ufl_operands[0].ufl_shape def __str__(self): return "%s^-1" % parstr(self.ufl_operands[0], self)
[docs]@ufl_type(is_index_free=True, num_ops=1) class Cofactor(CompoundTensorOperator): __slots__ = () def __init__(self, A): CompoundTensorOperator.__init__(self, (A,)) # Checks sh = A.ufl_shape if len(sh) != 2: error("Cofactor of tensor with rank != 2 is undefined.") if sh[0] != sh[1]: error("Cannot take cofactor of rectangular matrix with dimensions %s." % (sh,)) if A.ufl_free_indices: error("Not expecting free indices in Cofactor.") if isinstance(A, Zero): error("Cannot take cofactor of zero matrix.") @property def ufl_shape(self): return self.ufl_operands[0].ufl_shape def __str__(self): 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): __slots__ = () def __new__(cls, A): sh = A.ufl_shape # Checks if len(sh) != 2: error("Deviatoric part of tensor with rank != 2 is undefined.") if sh[0] != sh[1]: error("Cannot take deviatoric part of rectangular matrix with dimensions %s." % (sh,)) if A.ufl_free_indices: error("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): CompoundTensorOperator.__init__(self, (A,)) def __str__(self): 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): __slots__ = () def __new__(cls, A): sh = A.ufl_shape Afi = A.ufl_free_indices # Checks if len(sh) != 2: error("Skew symmetric part of tensor with rank != 2 is undefined.") if sh[0] != sh[1]: error("Cannot take skew part of rectangular matrix with dimensions %s." % (sh,)) if Afi: error("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): CompoundTensorOperator.__init__(self, (A,)) def __str__(self): 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): __slots__ = () def __new__(cls, A): sh = A.ufl_shape Afi = A.ufl_free_indices # Checks if len(sh) != 2: error("Symmetric part of tensor with rank != 2 is undefined.") if sh[0] != sh[1]: error("Cannot take symmetric part of rectangular matrix with dimensions %s." % (sh,)) if Afi: error("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): CompoundTensorOperator.__init__(self, (A,)) def __str__(self): return "sym(%s)" % self.ufl_operands[0]