Source code for pymatgen.core.tensors

# coding: utf-8
# Copyright (c) Pymatgen Development Team.
# Distributed under the terms of the MIT License.

"""
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
"""

import itertools
import warnings
import collections
import string
import os

import numpy as np

from scipy.linalg import polar

from monty.json import MSONable
from monty.serialization import loadfn

from pymatgen.symmetry.analyzer import SpacegroupAnalyzer
from pymatgen.core.operations import SymmOp
from pymatgen.core.lattice import Lattice
from pymatgen.analysis.structure_matcher import StructureMatcher

__author__ = "Joseph Montoya"
__copyright__ = "Copyright 2017, The Materials Project"
__credits__ = ("Maarten de Jong, Shyam Dwaraknath, Wei Chen, "
               "Mark Asta, Anubhav Jain, Terence Lew")
__version__ = "1.0"
__maintainer__ = "Joseph Montoya"
__email__ = "montoyjh@lbl.gov"
__status__ = "Production"
__date__ = "July 24, 2018"

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]])

DEFAULT_QUAD = loadfn(os.path.join(os.path.dirname(__file__),
                                   "quad_data.json"))


[docs]class Tensor(np.ndarray, MSONable): """ Base class for doing useful general operations on Nth order tensors, without restrictions on the type (stress, elastic, strain, piezo, etc.) """ symbol = "T" 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[()] 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 transform(self, symm_op): """ Applies a transformation (via a symmetry operation) to a tensor. Args: symm_op (SymmOp): a symmetry operation to apply to the tensor """ return self.__class__(symm_op.transform_tensor(self))
[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)
[docs] def project(self, n): """ Convenience method for projection of a tensor into a vector. Returns the tensor dotted into a unit vector along the input n. Args: n (3x1 array-like): direction to project onto Returns (float): scalar value corresponding to the projection of the tensor into the vector """ n = get_uvec(n) return self.einsum_sequence([n] * self.rank)
[docs] def average_over_unit_sphere(self, quad=None): """ Method for averaging the tensor projection over the unit with option for custom quadrature. Args: quad (dict): quadrature for integration, should be dictionary with "points" and "weights" keys defaults to quadpy.sphere.Lebedev(19) as read from file Returns: Average of tensor projected into vectors on the unit sphere """ quad = quad or DEFAULT_QUAD weights, points = quad['weights'], quad['points'] return sum([w * self.project(n) for w, n in zip(weights, points)])
[docs] def get_grouped_indices(self, voigt=False, **kwargs): """ Gets index sets for equivalent tensor values Args: voigt (bool): whether to get grouped indices of voigt or full notation tensor, defaults to false **kwargs: keyword args for np.isclose. Can take atol and rtol for absolute and relative tolerance, e. g. >>> tensor.group_array_indices(atol=1e-8) or >>> tensor.group_array_indices(rtol=1e-5) Returns: list of index groups where tensor values are equivalent to within tolerances """ if voigt: array = self.voigt else: array = self indices = list(itertools.product(*[range(n) for n in array.shape])) remaining = indices.copy() # Start with everything near zero grouped = [list(zip(*np.where(np.isclose(array, 0, **kwargs))))] remaining = [i for i in remaining if i not in grouped[0]] # Iteratively run through remaining indices while remaining: new = list(zip(*np.where(np.isclose( array, array[remaining[0]], **kwargs)))) grouped.append(new) remaining = [i for i in remaining if i not in new] # Don't return any empty lists return [g for g in grouped if g]
[docs] def get_symbol_dict(self, voigt=True, zero_index=False, **kwargs): """ Creates a summary dict for tensor with associated symbol Args: voigt (bool): whether to get symbol dict for voigt notation tensor, as opposed to full notation, defaults to true zero_index (bool): whether to set initial index to zero, defaults to false, since tensor notations tend to use one-indexing, rather than zero indexing like python **kwargs: keyword args for np.isclose. Can take atol and rtol for absolute and relative tolerance, e. g. >>> tensor.get_symbol_dict(atol=1e-8) or >>> tensor.get_symbol_dict(rtol=1e-5) Returns: list of index groups where tensor values are equivalent to within tolerances Returns: """ d = {} if voigt: array = self.voigt else: array = self grouped = self.get_grouped_indices(voigt=voigt, **kwargs) if zero_index: p = 0 else: p = 1 for indices in grouped: sym_string = self.symbol + '_' sym_string += ''.join([str(i + p) for i in indices[0]]) value = array[indices[0]] if not np.isclose(value, 0): d[sym_string] = array[indices[0]] return d
[docs] def round(self, decimals=0): """ Wrapper around numpy.round to ensure object of same type is returned Args: decimals :Number of decimal places to round to (default: 0). If decimals is negative, it specifies the number of positions to the left of the decimal point. Returns (Tensor): rounded tensor of same type """ return self.__class__(np.round(self, decimals=decimals))
@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, refine_rotation=True): """ 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 refine_rotation (bool): whether to refine the rotation using SquareTensor.refine_rotation """ # 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, x2 normal to ac plane elif xtal_sys == "triclinic": rotation = [vec / mag for (mag, vec) in sorted(zip(lengths, vecs))] rotation[1] = get_uvec(np.cross(rotation[2], rotation[0])) rotation[0] = np.cross(rotation[1], rotation[2]) rotation = SquareTensor(rotation) if refine_rotation: rotation = rotation.refine_rotation() return rotation
[docs] def convert_to_ieee(self, structure, initial_fit=True, refine_rotation=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. refine_rotation (bool): whether to refine the rotation produced by the ieee transform generator, default True """ rotation = self.get_ieee_rotation(structure, refine_rotation) result = self.copy() if initial_fit: result = result.fit_to_structure(structure) return result.rotate(rotation, tol=1e-2)
[docs] def structure_transform(self, original_structure, new_structure, refine_rotation=True): """ Transforms a tensor from one basis for an original structure into a new basis defined by a new structure. Args: original_structure (Structure): structure corresponding to the basis of the current tensor new_structure (Structure): structure corresponding to the desired basis refine_rotation (bool): whether to refine the rotations generated in get_ieee_rotation Returns: Tensor that has been transformed such that its basis corresponds to the new_structure's basis """ sm = StructureMatcher() if not sm.fit(original_structure, new_structure): warnings.warn("original and new structures do not match!") trans_1 = self.get_ieee_rotation(original_structure, refine_rotation) trans_2 = self.get_ieee_rotation(new_structure, refine_rotation) # Get the ieee format tensor new = self.rotate(trans_1) # Reverse the ieee format rotation for the second structure new = new.rotate(np.transpose(trans_2)) return new
[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") 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] def as_dict(self, voigt: bool = False) -> dict: """ Serializes the tensor object Args: voigt (bool): flag for whether to store entries in voigt-notation. Defaults to false, as information may be lost in conversion. Returns (Dict): serialized format tensor object """ input_array = self.voigt if voigt else self d = {"@module": self.__class__.__module__, "@class": self.__class__.__name__, "input_array": input_array.tolist()} if voigt: d.update({"voigt": voigt}) return d
[docs] @classmethod def from_dict(cls, d): """MSONAble from_dict implementation.""" voigt = d.get('voigt') if voigt: return cls.from_voigt(d["input_array"]) return cls(d["input_array"])
[docs]class TensorCollection(collections.abc.Sequence, MSONable): """ 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): """ :param tensor_list: List of tensors. :param base_class: Class to be used. """ 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): """ :param tol: Tolerance :return: TensorCollection where small values are set to 0. """ return self.__class__([t.zeroed(tol) for t in self])
[docs] def transform(self, symm_op): """ Transforms TensorCollection with a symmetry operation. :param symm_op: SymmetryOperation. :return: TensorCollection. """ return self.__class__([t.transform(symm_op) for t in self])
[docs] def rotate(self, matrix, tol=1e-3): """ Rotates TensorCollection. :param matrix: Rotation matrix. :param tol: tolerance. :return: TensorCollection. """ return self.__class__([t.rotate(matrix, tol) for t in self])
@property def symmetrized(self): """ :return: TensorCollection where all tensors are symmetrized. """ return self.__class__([t.symmetrized for t in self])
[docs] def is_symmetric(self, tol=1e-5): """ :param tol: tolerance :return: Whether all tensors are symmetric. """ return all([t.is_symmetric(tol) for t in self])
[docs] def fit_to_structure(self, structure, symprec=0.1): """ Fits all tensors to a Structure. :param structure: Structure :param symprec: symmetry precision. :return: TensorCollection. """ return self.__class__([t.fit_to_structure(structure, symprec) for t in self])
[docs] def is_fit_to_structure(self, structure, tol=1e-2): """ :param structure: Structure :param tol: tolerance :return: Whether all tensors are fitted to Structure. """ return all([t.is_fit_to_structure(structure, tol) for t in self])
@property def voigt(self): """ :return: TensorCollection where all tensors are in voight form. """ return [t.voigt for t in self] @property def ranks(self): """ :return: Ranks for all tensors. """ return [t.rank for t in self]
[docs] def is_voigt_symmetric(self, tol=1e-6): """ :param tol: tolerance :return: Whether all tensors are voigt symmetric. """ return all([t.is_voigt_symmetric(tol) for t in self])
[docs] @classmethod def from_voigt(cls, voigt_input_list, base_class=Tensor): """ Creates TensorCollection from voigt form. :param voigt_input_list: List of voigt tensors :param base_class: Class for tensor. :return: TensorCollection. """ return cls([base_class.from_voigt(v) for v in voigt_input_list])
[docs] def convert_to_ieee(self, structure, initial_fit=True, refine_rotation=True): """ Convert all tensors to IEEE. :param structure: Structure :param initial_fit: Whether to perform an initial fit. :param refine_rotation: Whether to refine the rotation. :return: TensorCollection. """ return self.__class__( [t.convert_to_ieee(structure, initial_fit, refine_rotation) for t in self])
[docs] def round(self, *args, **kwargs): """ Round all tensors. :param args: Passthrough to Tensor.round :param kwargs: Passthrough to Tensor.round :return: TensorCollection. """ return self.__class__([t.round(*args, **kwargs) for t in self])
@property def voigt_symmetrized(self): """ :return: TensorCollection where all tensors are voigt symmetrized. """ return self.__class__([t.voigt_symmetrized for t in self])
[docs] def as_dict(self, voigt=False): """ :param voigt: Whether to use voight form. :return: Dict representation of TensorCollection. """ tensor_list = self.voigt if voigt else self d = {"@module": self.__class__.__module__, "@class": self.__class__.__name__, "tensor_list": [t.tolist() for t in tensor_list]} if voigt: d.update({"voigt": voigt}) return d
[docs] @classmethod def from_dict(cls, d): """ Creates TensorCollection from dict. :param d: dict :return: TensorCollection """ voigt = d.get('voigt') if voigt: return cls.from_voigt(d["tensor_list"]) return cls(d["tensor_list"])
[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().__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 refine_rotation(self): """ Helper method for refining rotation matrix by ensuring that second and third rows are perpindicular to the first. Gets new y vector from an orthogonal projection of x onto y and the new z vector from a cross product of the new x and y Args: tol to test for rotation Returns: new rotation matrix """ new_x, y = get_uvec(self[0]), get_uvec(self[1]) # Get a projection on y new_y = y - np.dot(new_x, y) * new_x new_z = np.cross(new_x, new_y) return SquareTensor([new_x, new_y, new_z])
[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_mapping = TensorMapping([tensors[0]], [[]], tol=tol) for tensor in tensors[1:]: is_unique = True for unique_tensor, symmop in itertools.product(unique_mapping, symmops): if np.allclose(unique_tensor.transform(symmop), tensor, atol=tol): unique_mapping[unique_tensor].append(symmop) is_unique = False break if is_unique: unique_mapping[tensor] = [] return unique_mapping
[docs]class TensorMapping(collections.abc.MutableMapping): """ Base class for tensor mappings, which function much like a dictionary, but use numpy routines to determine approximate equality to keys for getting and setting items. This is intended primarily for convenience with things like stress-strain pairs and fitting data manipulation. In general, it is significantly less robust than a typical hashing and should be used with care. """ def __init__(self, tensors=None, values=None, tol=1e-5): """ Initialize a TensorMapping Args: tensor_list ([Tensor]): list of tensors value_list ([]): list of values to be associated with tensors tol (float): an absolute tolerance for getting and setting items in the mapping """ self._tensor_list = tensors or [] self._value_list = values or [] if not len(self._tensor_list) == len(self._value_list): raise ValueError("TensorMapping must be initialized with tensors" "and values of equivalent length") self.tol = tol def __getitem__(self, item): index = self._get_item_index(item) if index is None: raise KeyError("{} not found in mapping.".format(item)) return self._value_list[index] def __setitem__(self, key, value): index = self._get_item_index(key) if index is None: self._tensor_list.append(key) self._value_list.append(value) else: self._value_list[index] = value def __delitem__(self, key): index = self._get_item_index(key) self._tensor_list.pop(index) self._value_list.pop(index) def __len__(self): return len(self._tensor_list) def __iter__(self): for item in self._tensor_list: yield item
[docs] def values(self): """ :return: Values in mapping. """ return self._value_list
[docs] def items(self): """ :return: Items in mapping. """ return zip(self._tensor_list, self._value_list)
def __contains__(self, item): return not self._get_item_index(item) is None def _get_item_index(self, item): if len(self._tensor_list) == 0: return None item = np.array(item) axis = tuple(range(1, len(item.shape) + 1)) mask = np.all(np.abs(np.array(self._tensor_list) - item) < self.tol, axis=axis) indices = np.where(mask)[0] if len(indices) > 1: raise ValueError("Tensor key collision.") if len(indices) == 0: return None return indices[0]