pymatgen.analysis.elasticity.tensors module¶

class
SquareTensor
[source]¶ Bases:
pymatgen.analysis.elasticity.tensors.Tensor
Base class for doing useful general operations on second rank tensors (stress, strain etc.).
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 nonsquare matrix.
Parameters:  input_array (3x3 arraylike) – the 3x3 arraylike representing the content of the tensor
 vscale (6x1 arraylike) – 6x1 arraylike scaling the voigtnotation vector with the tensor entries

det
¶ shorthand for the determinant of the SquareTensor

get_scaled
(scale_factor)[source]¶ Scales the tensor by a certain multiplicative scale factor
Parameters: scale_factor (float) – scalar multiplier to be applied to the SquareTensor object

inv
¶ shorthand for matrix inverse on SquareTensor

is_rotation
(tol=0.001, include_improper=True)[source]¶ 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
Parameters:  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

principal_invariants
¶ Returns a list of principal invariants for the tensor, which are the values of the coefficients of the characteristic polynomial for the matrix

refine_rotation
()[source]¶ 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
Parameters: to test for rotation (tol) – Returns: new rotation matrix

trans
¶ shorthand for transpose on SquareTensor

class
Tensor
[source]¶ Bases:
numpy.ndarray
Base class for doing useful general operations on Nth order tensors, without restrictions on the type (stress, elastic, strain, piezo, etc.)
Create a Tensor object. Note that the constructor uses __new__ rather than __init__ according to the standard method of subclassing numpy ndarrays.
Parameters:  input_array – (arraylike with shape 3^N): arraylike representing a tensor quantity in standard (i. e. nonvoigt) notation
 vscale – (N x M arraylike): a matrix corresponding to the coefficients of the voigtnotation tensor

convert_to_ieee
(structure, initial_fit=True, refine_rotation=True)[source]¶ Given a structure associated with a tensor, attempts a calculation of the tensor in IEEE format according to the 1987 IEEE standards.
Parameters:  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

einsum_sequence
(other_arrays, einsum_string=None)[source]¶ Calculates the result of an einstein summation expression

fit_to_structure
(structure, symprec=0.1)[source]¶ Returns a tensor that is invariant with respect to symmetry operations corresponding to a structure
Parameters:  structure (Structure) – structure from which to generate symmetry operations
 symprec (float) – symmetry tolerance for the Spacegroup Analyzer used to generate the symmetry operations

classmethod
from_values_indices
(values, indices, populate=False, structure=None, voigt_rank=None, vsym=True, verbose=False)[source]¶ Creates a tensor from values and indices, with options for populating the remainder of the tensor.
Parameters:  values (floats) – numbers to place at indices
 indices (arraylikes) – 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

classmethod
from_voigt
(voigt_input)[source]¶ Constructor based on the voigt notation vector or matrix.
Parameters: voigt_input (arraylike) – voigt input for a given tensor

static
get_ieee_rotation
(structure, refine_rotation=True)[source]¶ Given a structure associated with a tensor, determines the rotation matrix for IEEE conversion according to the 1987 IEEE standards.
Parameters:  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

static
get_voigt_dict
(rank)[source]¶ Returns a dictionary that maps indices in the tensor to those in a voigt representation based on input rank
Parameters: rank (int) – Tensor rank to generate the voigt map

is_fit_to_structure
(structure, tol=0.01)[source]¶ 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
Parameters:  structure (Structure) – structure to be fit to
 tol (float) – tolerance for symmetry testing

is_symmetric
(tol=1e05)[source]¶ Tests whether a tensor is symmetric or not based on the residual with its symmetric part, from self.symmetrized
Parameters: tol (float) – tolerance to test for symmetry

is_voigt_symmetric
(tol=1e06)[source]¶ Tests symmetry of tensor to that necessary for voigtconversion by grouping indices into pairs and constructing a sequence of possible permutations to be used in a tensor transpose

populate
(structure, prec=1e05, maxiter=200, verbose=False, precond=True, vsym=True)[source]¶ Takes a partially populated tensor, and populates the nonzero entries according to the following procedure, iterated until the desired convergence (specified via prec) is achieved.
 Find nonzero entries
 Symmetrize the tensor with respect to crystal symmetry and (optionally) voigt symmetry
 Reset the nonzero entries of the original tensor
Parameters:  structure (structure object) –
 prec (float) – precision for determining a nonzero 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

rotate
(matrix, tol=0.001)[source]¶ Applies a rotation directly, and tests input matrix to ensure a valid rotation.
Parameters:  matrix (3x3 arraylike) – rotation matrix to be applied to tensor
 tol (float) – tolerance for testing rotation matrix validity

structure_transform
(original_structure, new_structure, refine_rotation=True)[source]¶ Transforms a tensor from one basis for an original structure into a new basis defined by a new structure.
Parameters: Returns: Tensor that has been transformed such that its basis corresponds to the new_structure’s basis

symmetrized
¶ Returns a generally symmetrized tensor, calculated by taking the sum of the tensor and its transpose with respect to all possible permutations of indices

transform
(symm_op)[source]¶ Applies a transformation (via a symmetry operation) to a tensor.
Parameters: symm_op (SymmOp) – a symmetry operation to apply to the tensor

voigt
¶ Returns the tensor in Voigt notation

voigt_symmetrized
¶ Returns a “voigt”symmetrized tensor, i. e. a voigtnotation tensor such that it is invariant wrt permutation of indices

class
TensorCollection
(tensor_list, base_class=<class 'pymatgen.analysis.elasticity.tensors.Tensor'>)[source]¶ Bases:
collections.abc.Sequence
A sequence of tensors that can be used for fitting data or for having a tensor expansion

classmethod
from_voigt
(voigt_input_list, base_class=<class 'pymatgen.analysis.elasticity.tensors.Tensor'>)[source]¶

ranks
¶

symmetrized
¶

voigt
¶

classmethod

get_tkd_value
(tensor_keyed_dict, tensor, allclose_kwargs=None)[source]¶ Helper function to find a value in a tensorkeyed dictionary using an approximation to the key. This is useful if rounding errors in construction occur or hashing issues arise in tensorkeyeddictionaries (e. g. from symmetry_reduce). Resolves most hashing issues, and is preferable to redefining eq methods in the base tensor class.
Parameters:  tensor_keyed_dict (dict) – dict with Tensor keys
 tensor (Tensor) – tensor to find value of in the dict
 allclose_kwargs (dict) – dict of keywordargs to pass to allclose.

symmetry_reduce
(tensors, structure, tol=1e08, **kwargs)[source]¶ 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
Parameters:  tensors (list of tensors) – list of Tensor objects to test for symmetricallyequivalent 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