# coding: utf-8
# Copyright (c) Pymatgen Development Team.
# Distributed under the terms of the MIT License.
from __future__ import division, print_function, unicode_literals, \
absolute_import
from scipy.linalg import polar
import numpy as np
import itertools
import warnings
import collections
import string
from pymatgen.symmetry.analyzer import SpacegroupAnalyzer
from pymatgen.core.operations import SymmOp
from pymatgen.core.lattice import Lattice
"""
This module provides a base class for tensor-like objects and methods for
basic tensor manipulation. It also provides a class, SquareTensor,
that provides basic methods for creating and manipulating rank 2 tensors
"""
__author__ = "Maarten de Jong"
__copyright__ = "Copyright 2012, The Materials Project"
__credits__ = ("Joseph Montoya, Shyam Dwaraknath, Wei Chen, "
"Mark Asta, Anubhav Jain, Terence Lew")
__version__ = "1.0"
__maintainer__ = "Joseph Montoya"
__email__ = "montoyjh@lbl.gov"
__status__ = "Development"
__date__ = "March 22, 2012"
voigt_map = [(0, 0), (1, 1), (2, 2), (1, 2), (0, 2), (0, 1)]
reverse_voigt_map = np.array([[0, 5, 4],
[5, 1, 3],
[4, 3, 2]])
[docs]class Tensor(np.ndarray):
"""
Base class for doing useful general operations on Nth order tensors,
without restrictions on the type (stress, elastic, strain, piezo, etc.)
"""
def __new__(cls, input_array, vscale=None, check_rank=None):
"""
Create a Tensor object. Note that the constructor uses __new__
rather than __init__ according to the standard method of
subclassing numpy ndarrays.
Args:
input_array: (array-like with shape 3^N): array-like representing
a tensor quantity in standard (i. e. non-voigt) notation
vscale: (N x M array-like): a matrix corresponding
to the coefficients of the voigt-notation tensor
"""
obj = np.asarray(input_array).view(cls)
obj.rank = len(obj.shape)
if check_rank and check_rank != obj.rank:
raise ValueError("{} input must be rank {}".format(
obj.__class__.__name__, check_rank))
vshape = tuple([3] * (obj.rank % 2) + [6] * (obj.rank // 2))
obj._vscale = np.ones(vshape)
if vscale is not None:
obj._vscale = vscale
if obj._vscale.shape != vshape:
raise ValueError("Voigt scaling matrix must be the shape of the "
"voigt notation matrix or vector.")
if not all([i == 3 for i in obj.shape]):
raise ValueError("Pymatgen only supports 3-dimensional tensors, "
"and default tensor constructor uses standard "
"notation. To construct from voigt notation, use"
" {}.from_voigt".format(obj.__class__.__name__))
return obj
def __array_finalize__(self, obj):
if obj is None:
return
self.rank = getattr(obj, 'rank', None)
self._vscale = getattr(obj, '_vscale', None)
self._vdict = getattr(obj, '_vdict', None)
def __array_wrap__(self, obj):
"""
Overrides __array_wrap__ methods in ndarray superclass to avoid errors
associated with functions that return scalar values
"""
if len(obj.shape) == 0:
return obj[()]
else:
return np.ndarray.__array_wrap__(self, obj)
def __hash__(self):
"""
define a hash function, since numpy arrays
have their own __eq__ method
"""
return hash(self.tostring())
def __repr__(self):
return "{}({})".format(self.__class__.__name__,
self.__str__())
[docs] def zeroed(self, tol=1e-3):
"""
returns the matrix with all entries below a certain threshold
(i.e. tol) set to zero
"""
new_tensor = self.copy()
new_tensor[abs(new_tensor) < tol] = 0
return new_tensor
[docs] def rotate(self, matrix, tol=1e-3):
"""
Applies a rotation directly, and tests input matrix to ensure a valid
rotation.
Args:
matrix (3x3 array-like): rotation matrix to be applied to tensor
tol (float): tolerance for testing rotation matrix validity
"""
matrix = SquareTensor(matrix)
if not matrix.is_rotation(tol):
raise ValueError("Rotation matrix is not valid.")
sop = SymmOp.from_rotation_and_translation(matrix,
[0., 0., 0.])
return self.transform(sop)
[docs] def einsum_sequence(self, other_arrays, einsum_string=None):
"""
Calculates the result of an einstein summation expression
"""
if not isinstance(other_arrays, list):
raise ValueError("other tensors must be list of "
"tensors or tensor input")
other_arrays = [np.array(a) for a in other_arrays]
if not einsum_string:
lc = string.ascii_lowercase
einsum_string = lc[:self.rank]
other_ranks = [len(a.shape) for a in other_arrays]
idx = self.rank - sum(other_ranks)
for length in other_ranks:
einsum_string += ',' + lc[idx:idx + length]
idx += length
einsum_args = [self] + list(other_arrays)
return np.einsum(einsum_string, *einsum_args)
@property
def symmetrized(self):
"""
Returns a generally symmetrized tensor, calculated by taking
the sum of the tensor and its transpose with respect to all
possible permutations of indices
"""
perms = list(itertools.permutations(range(self.rank)))
return sum([np.transpose(self, ind) for ind in perms]) / len(perms)
@property
def voigt_symmetrized(self):
"""
Returns a "voigt"-symmetrized tensor, i. e. a voigt-notation
tensor such that it is invariant wrt permutation of indices
"""
if not (self.rank % 2 == 0 and self.rank > 2):
raise ValueError("V-symmetrization requires rank even and > 2")
v = self.voigt
perms = list(itertools.permutations(range(len(v.shape))))
new_v = sum([np.transpose(v, ind) for ind in perms]) / len(perms)
return self.__class__.from_voigt(new_v)
[docs] def is_symmetric(self, tol=1e-5):
"""
Tests whether a tensor is symmetric or not based on the residual
with its symmetric part, from self.symmetrized
Args:
tol (float): tolerance to test for symmetry
"""
return (self - self.symmetrized < tol).all()
[docs] def fit_to_structure(self, structure, symprec=0.1):
"""
Returns a tensor that is invariant with respect to symmetry
operations corresponding to a structure
Args:
structure (Structure): structure from which to generate
symmetry operations
symprec (float): symmetry tolerance for the Spacegroup Analyzer
used to generate the symmetry operations
"""
sga = SpacegroupAnalyzer(structure, symprec)
symm_ops = sga.get_symmetry_operations(cartesian=True)
return sum([self.transform(symm_op)
for symm_op in symm_ops]) / len(symm_ops)
[docs] def is_fit_to_structure(self, structure, tol=1e-2):
"""
Tests whether a tensor is invariant with respect to the
symmetry operations of a particular structure by testing
whether the residual of the symmetric portion is below a
tolerance
Args:
structure (Structure): structure to be fit to
tol (float): tolerance for symmetry testing
"""
return (self - self.fit_to_structure(structure) < tol).all()
@property
def voigt(self):
"""
Returns the tensor in Voigt notation
"""
v_matrix = np.zeros(self._vscale.shape, dtype=self.dtype)
this_voigt_map = self.get_voigt_dict(self.rank)
for ind in this_voigt_map:
v_matrix[this_voigt_map[ind]] = self[ind]
if not self.is_voigt_symmetric():
warnings.warn("Tensor is not symmetric, information may "
"be lost in voigt conversion.")
return v_matrix * self._vscale
[docs] def is_voigt_symmetric(self, tol=1e-6):
"""
Tests symmetry of tensor to that necessary for voigt-conversion
by grouping indices into pairs and constructing a sequence of
possible permutations to be used in a tensor transpose
"""
transpose_pieces = [[[0 for i in range(self.rank % 2)]]]
transpose_pieces += [[range(j, j + 2)] for j in
range(self.rank % 2, self.rank, 2)]
for n in range(self.rank % 2, len(transpose_pieces)):
if len(transpose_pieces[n][0]) == 2:
transpose_pieces[n] += [transpose_pieces[n][0][::-1]]
for trans_seq in itertools.product(*transpose_pieces):
trans_seq = list(itertools.chain(*trans_seq))
if (self - self.transpose(trans_seq) > tol).any():
return False
return True
[docs] @staticmethod
def get_voigt_dict(rank):
"""
Returns a dictionary that maps indices in the tensor to those
in a voigt representation based on input rank
Args:
rank (int): Tensor rank to generate the voigt map
"""
vdict = {}
for ind in itertools.product(*[range(3)] * rank):
v_ind = ind[:rank % 2]
for j in range(rank // 2):
pos = rank % 2 + 2 * j
v_ind += (reverse_voigt_map[ind[pos:pos + 2]],)
vdict[ind] = v_ind
return vdict
[docs] @classmethod
def from_voigt(cls, voigt_input):
"""
Constructor based on the voigt notation vector or matrix.
Args:
voigt_input (array-like): voigt input for a given tensor
"""
voigt_input = np.array(voigt_input)
rank = sum(voigt_input.shape) // 3
t = cls(np.zeros([3] * rank))
if voigt_input.shape != t._vscale.shape:
raise ValueError("Invalid shape for voigt matrix")
voigt_input = voigt_input / t._vscale
this_voigt_map = t.get_voigt_dict(rank)
for ind in this_voigt_map:
t[ind] = voigt_input[this_voigt_map[ind]]
return cls(t)
[docs] @staticmethod
def get_ieee_rotation(structure):
"""
Given a structure associated with a tensor, determines
the rotation matrix for IEEE conversion according to
the 1987 IEEE standards.
Args:
structure (Structure): a structure associated with the
tensor to be converted to the IEEE standard
"""
# Check conventional setting:
sga = SpacegroupAnalyzer(structure)
dataset = sga.get_symmetry_dataset()
trans_mat = dataset['transformation_matrix']
conv_latt = Lattice(np.transpose(np.dot(np.transpose(
structure.lattice.matrix), np.linalg.inv(trans_mat))))
xtal_sys = sga.get_crystal_system()
vecs = conv_latt.matrix
lengths = np.array(conv_latt.abc)
angles = np.array(conv_latt.angles)
rotation = np.zeros((3, 3))
# IEEE rules: a,b,c || x1,x2,x3
if xtal_sys == "cubic":
rotation = [vecs[i] / lengths[i] for i in range(3)]
# IEEE rules: a=b in length; c,a || x3, x1
elif xtal_sys == "tetragonal":
rotation = np.array([vec / mag for (mag, vec) in
sorted(zip(lengths, vecs),
key=lambda x: x[0])])
if abs(lengths[2] - lengths[1]) < abs(lengths[1] - lengths[0]):
rotation[0], rotation[2] = rotation[2], rotation[0].copy()
rotation[1] = get_uvec(np.cross(rotation[2], rotation[0]))
# IEEE rules: c<a<b; c,a || x3,x1
elif xtal_sys == "orthorhombic":
rotation = [vec / mag for (mag, vec) in sorted(zip(lengths, vecs))]
rotation = np.roll(rotation, 2, axis=0)
# IEEE rules: c,a || x3,x1, c is threefold axis
# Note this also includes rhombohedral crystal systems
elif xtal_sys in ("trigonal", "hexagonal"):
# find threefold axis:
tf_index = np.argmin(abs(angles - 120.))
non_tf_mask = np.logical_not(angles == angles[tf_index])
rotation[2] = get_uvec(vecs[tf_index])
rotation[0] = get_uvec(vecs[non_tf_mask][0])
rotation[1] = get_uvec(np.cross(rotation[2], rotation[0]))
# IEEE rules: b,c || x2,x3; alpha=beta=90, c<a
elif xtal_sys == "monoclinic":
# Find unique axis
u_index = np.argmax(abs(angles - 90.))
n_umask = np.logical_not(angles == angles[u_index])
rotation[1] = get_uvec(vecs[u_index])
# Shorter of remaining lattice vectors for c axis
c = [vec / mag for (mag, vec) in
sorted(zip(lengths[n_umask], vecs[n_umask]))][0]
rotation[2] = np.array(c)
rotation[0] = np.cross(rotation[1], rotation[2])
# IEEE rules: c || x3
elif xtal_sys == "triclinic":
rotation = [vec / mag for (mag, vec) in sorted(zip(lengths, vecs))]
rotation = np.roll(rotation, 2, axis=0)
rotation[1] = get_uvec(np.cross(rotation[2], rotation[1]))
rotation[0] = np.cross(rotation[1], rotation[2])
return rotation
[docs] def convert_to_ieee(self, structure, initial_fit=True):
"""
Given a structure associated with a tensor, attempts a
calculation of the tensor in IEEE format according to
the 1987 IEEE standards.
Args:
structure (Structure): a structure associated with the
tensor to be converted to the IEEE standard
initial_fit (bool): flag to indicate whether initial
tensor is fit to the symmetry of the structure.
Defaults to true. Note that if false, inconsistent
results may be obtained due to symmetrically
equivalent, but distinct transformations
being used in different versions of spglib.
"""
rotation = self.get_ieee_rotation(structure)
result = self.copy()
if initial_fit:
result = result.fit_to_structure(structure)
return result.rotate(rotation, tol=1e-2)
[docs] @classmethod
def from_values_indices(cls, values, indices, populate=False,
structure=None, voigt_rank=None,
vsym=True, verbose=False):
"""
Creates a tensor from values and indices, with options
for populating the remainder of the tensor.
Args:
values (floats): numbers to place at indices
indices (array-likes): indices to place values at
populate (bool): whether to populate the tensor
structure (Structure): structure to base population
or fit_to_structure on
voigt_rank (int): full tensor rank to indicate the
shape of the resulting tensor. This is necessary
if one provides a set of indices more minimal than
the shape of the tensor they want, e.g.
Tensor.from_values_indices((0, 0), 100)
vsym (bool): whether to voigt symmetrize during the
optimization procedure
verbose (bool): whether to populate verbosely
"""
# auto-detect voigt notation
# TODO: refactor rank inheritance to make this easier
indices = np.array(indices)
if voigt_rank:
shape = ([3]*(voigt_rank % 2) + [6]*(voigt_rank // 2))
else:
shape = np.ceil(np.max(indices+1, axis=0) / 3.) * 3
base = np.zeros(shape.astype(int))
for v, idx in zip(values, indices):
base[tuple(idx)] = v
if 6 in shape:
obj = cls.from_voigt(base)
else:
obj = cls(base)
if populate:
assert structure, "Populate option must include structure input"
obj = obj.populate(structure, vsym=vsym, verbose=verbose)
elif structure:
obj = obj.fit_to_structure(structure)
return obj
[docs] def populate(self, structure, prec=1e-5, maxiter=200, verbose=False,
precond=True, vsym=True):
"""
Takes a partially populated tensor, and populates the non-zero
entries according to the following procedure, iterated until
the desired convergence (specified via prec) is achieved.
1. Find non-zero entries
2. Symmetrize the tensor with respect to crystal symmetry and
(optionally) voigt symmetry
3. Reset the non-zero entries of the original tensor
Args:
structure (structure object)
prec (float): precision for determining a non-zero value
maxiter (int): maximum iterations for populating the tensor
verbose (bool): whether to populate verbosely
precond (bool): whether to precondition by cycling through
all symmops and storing new nonzero values, default True
vsym (bool): whether to enforce voigt symmetry, defaults
to True
"""
if precond:
# Generate the guess from populated
sops = SpacegroupAnalyzer(structure).get_symmetry_operations()
guess = Tensor(np.zeros(self.shape))
mask = abs(self) > prec
guess[mask] = self[mask]
def merge(old, new):
gmask = np.abs(old) > prec
nmask = np.abs(new) > prec
new_mask = np.logical_not(gmask)*nmask
avg_mask = gmask*nmask
old[avg_mask] = (old[avg_mask] + new[avg_mask]) / 2.
old[new_mask] = new[new_mask]
if verbose:
print("Preconditioning for {} symmops".format(len(sops)))
for sop in sops:
rot = guess.transform(sop)
# Store non-zero entries of new that weren't previously
# in the guess in the guess
merge(guess, rot)
if verbose:
print("Preconditioning for voigt symmetry".format(len(sops)))
if vsym:
v = guess.voigt
perms = list(itertools.permutations(range(len(v.shape))))
for perm in perms:
vtrans = np.transpose(v, perm)
merge(v, vtrans)
guess = Tensor.from_voigt(v)
else:
guess = np.zeros(self.shape)
assert guess.shape == self.shape, "Guess must have same shape"
converged = False
test_new, test_old = [guess.copy()]*2
for i in range(maxiter):
test_new = test_old.fit_to_structure(structure)
if vsym:
test_new = test_new.voigt_symmetrized
diff = np.abs(test_old - test_new)
converged = (diff < prec).all()
if converged:
break
test_new[mask] = self[mask]
test_old = test_new
if verbose:
print("Iteration {}: {}".format(i, np.max(diff)))
if not converged:
max_diff = np.max(np.abs(self - test_new))
warnings.warn("Warning, populated tensor is not converged "
"with max diff of {}".format(max_diff))
return self.__class__(test_new)
[docs]class TensorCollection(collections.Sequence):
"""
A sequence of tensors that can be used for fitting data
or for having a tensor expansion
"""
def __init__(self, tensor_list, base_class=Tensor):
self.tensors = [base_class(t) if not isinstance(t, base_class)
else t for t in tensor_list]
def __len__(self):
return len(self.tensors)
def __getitem__(self, ind):
return self.tensors[ind]
def __iter__(self):
return self.tensors.__iter__()
[docs] def zeroed(self, tol=1e-3):
return self.__class__([t.zeroed(tol) for t in self])
[docs] def rotate(self, matrix, tol=1e-3):
return self.__class__([t.rotate(matrix, tol) for t in self])
@property
def symmetrized(self):
return self.__class__([t.symmetrized for t in self])
[docs] def is_symmetric(self, tol=1e-5):
return all([t.is_symmetric(tol) for t in self])
[docs] def fit_to_structure(self, structure, symprec=0.1):
return self.__class__([t.fit_to_structure(structure, symprec)
for t in self])
[docs] def is_fit_to_structure(self, structure, tol=1e-2):
return all([t.is_fit_to_structure(structure, tol) for t in self])
@property
def voigt(self):
return [t.voigt for t in self]
@property
def ranks(self):
return [t.rank for t in self]
[docs] def is_voigt_symmetric(self, tol=1e-6):
return all([t.is_voigt_symmetric(tol) for t in self])
[docs] @classmethod
def from_voigt(cls, voigt_input_list, base_class=Tensor):
return cls([base_class.from_voigt(v) for v in voigt_input_list])
[docs] def convert_to_ieee(self, structure):
return self.__class__([t.convert_to_ieee(structure) for t in self])
[docs]class SquareTensor(Tensor):
"""
Base class for doing useful general operations on second rank tensors
(stress, strain etc.).
"""
def __new__(cls, input_array, vscale=None):
"""
Create a SquareTensor object. Note that the constructor uses __new__
rather than __init__ according to the standard method of
subclassing numpy ndarrays. Error is thrown when the class is
initialized with non-square matrix.
Args:
input_array (3x3 array-like): the 3x3 array-like
representing the content of the tensor
vscale (6x1 array-like): 6x1 array-like scaling the
voigt-notation vector with the tensor entries
"""
obj = super(SquareTensor, cls).__new__(cls, input_array, vscale,
check_rank=2)
return obj.view(cls)
@property
def trans(self):
"""
shorthand for transpose on SquareTensor
"""
return SquareTensor(np.transpose(self))
@property
def inv(self):
"""
shorthand for matrix inverse on SquareTensor
"""
if self.det == 0:
raise ValueError("SquareTensor is non-invertible")
return SquareTensor(np.linalg.inv(self))
@property
def det(self):
"""
shorthand for the determinant of the SquareTensor
"""
return np.linalg.det(self)
[docs] def is_rotation(self, tol=1e-3, include_improper=True):
"""
Test to see if tensor is a valid rotation matrix, performs a
test to check whether the inverse is equal to the transpose
and if the determinant is equal to one within the specified
tolerance
Args:
tol (float): tolerance to both tests of whether the
the determinant is one and the inverse is equal
to the transpose
include_improper (bool): whether to include improper
rotations in the determination of validity
"""
det = np.abs(np.linalg.det(self))
if include_improper:
det = np.abs(det)
return (np.abs(self.inv - self.trans) < tol).all() \
and (np.abs(det - 1.) < tol)
[docs] def get_scaled(self, scale_factor):
"""
Scales the tensor by a certain multiplicative scale factor
Args:
scale_factor (float): scalar multiplier to be applied to the
SquareTensor object
"""
return SquareTensor(self * scale_factor)
@property
def principal_invariants(self):
"""
Returns a list of principal invariants for the tensor,
which are the values of the coefficients of the characteristic
polynomial for the matrix
"""
return np.poly(self)[1:] * np.array([-1, 1, -1])
[docs] def polar_decomposition(self, side='right'):
"""
calculates matrices for polar decomposition
"""
return polar(self, side=side)
[docs]def get_uvec(vec):
""" Gets a unit vector parallel to input vector"""
l = np.linalg.norm(vec)
if l < 1e-8:
return vec
return vec / l
[docs]def symmetry_reduce(tensors, structure, tol=1e-8, **kwargs):
"""
Function that converts a list of tensors corresponding to a structure
and returns a dictionary consisting of unique tensor keys with symmop
values corresponding to transformations that will result in derivative
tensors from the original list
Args:
tensors (list of tensors): list of Tensor objects to test for
symmetrically-equivalent duplicates
structure (Structure): structure from which to get symmetry
tol (float): tolerance for tensor equivalence
kwargs: keyword arguments for the SpacegroupAnalyzer
returns:
dictionary consisting of unique tensors with symmetry operations
corresponding to those which will reconstruct the remaining
tensors as values
"""
sga = SpacegroupAnalyzer(structure, **kwargs)
symmops = sga.get_symmetry_operations(cartesian=True)
unique_tdict = {}
for tensor in tensors:
is_unique = True
for unique_tensor, symmop in itertools.product(unique_tdict, symmops):
if (np.abs(unique_tensor.transform(symmop) - tensor) < tol).all():
unique_tdict[unique_tensor].append(symmop)
is_unique = False
break
if is_unique:
unique_tdict[tensor] = []
return unique_tdict
[docs]def get_tkd_value(tensor_keyed_dict, tensor, allclose_kwargs={}):
"""
Helper function to find a value in a tensor-keyed-
dictionary using an approximation to the key. This
is useful if rounding errors in construction occur
or hashing issues arise in tensor-keyed-dictionaries
(e. g. from symmetry_reduce). Resolves most
hashing issues, and is preferable to redefining
eq methods in the base tensor class.
Args:
tensor_keyed_dict (dict): dict with Tensor keys
tensor (Tensor): tensor to find value of in the dict
allclose_kwargs (dict): dict of keyword-args
to pass to allclose.
"""
for tkey, value in tensor_keyed_dict.items():
if np.allclose(tensor, tkey, **allclose_kwargs):
return value