919 lines
37 KiB
Python
919 lines
37 KiB
Python
import inspect
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import copy
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import numpy as np
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from . import Rotation
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from . import Crystal
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from . import util
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from . import tensor
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_parameter_doc = \
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"""
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family : {'triclinic', 'monoclinic', 'orthorhombic', 'tetragonal', 'hexagonal', 'cubic'}, optional.
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Name of the crystal family.
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Family will be inferred if 'lattice' is given.
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lattice : {'aP', 'mP', 'mS', 'oP', 'oS', 'oI', 'oF', 'tP', 'tI', 'hP', 'cP', 'cI', 'cF'}, optional.
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Name of the Bravais lattice in Pearson notation.
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a : float, optional
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Length of lattice parameter 'a'.
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b : float, optional
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Length of lattice parameter 'b'.
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c : float, optional
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Length of lattice parameter 'c'.
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alpha : float, optional
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Angle between b and c lattice basis.
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beta : float, optional
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Angle between c and a lattice basis.
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gamma : float, optional
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Angle between a and b lattice basis.
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degrees : bool, optional
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Angles are given in degrees. Defaults to False.
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"""
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class Orientation(Rotation,Crystal):
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"""
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Representation of crystallographic orientation as combination of rotation and either crystal family or Bravais lattice.
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The crystal family is one of:
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- triclinic
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- monoclinic
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- orthorhombic
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- tetragonal
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- hexagonal
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- cubic
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and enables symmetry-related operations such as
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"equivalent", "reduced", "disorientation", "IPF_color", or "to_SST".
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The Bravais lattice is given in the Pearson notation:
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- triclinic
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- aP : primitive
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- monoclinic
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- mP : primitive
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- mS : base-centered
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- orthorhombic
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- oP : primitive
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- oS : base-centered
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- oI : body-centered
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- oF : face-centered
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- tetragonal
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- tP : primitive
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- tI : body-centered
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- hexagonal
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- hP : primitive
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- cubic
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- cP : primitive
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- cI : body-centered
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- cF : face-centered
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and inherits the corresponding crystal family.
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Specifying a Bravais lattice, compared to just the crystal family,
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extends the functionality of Orientation objects to include operations such as
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"Schmid", "related", or "to_pole" that require a lattice type and its parameters.
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Examples
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--------
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An array of 3 x 5 random orientations reduced to the fundamental zone of tetragonal symmetry:
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>>> import damask
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>>> o=damask.Orientation.from_random(shape=(3,5),family='tetragonal').reduced
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"""
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@util.extend_docstring(_parameter_doc)
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def __init__(self,
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rotation = np.array([1.0,0.0,0.0,0.0]), *,
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family = None,
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lattice = None,
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a = None,b = None,c = None,
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alpha = None,beta = None,gamma = None,
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degrees = False):
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"""
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New orientation.
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Parameters
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----------
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rotation : list, numpy.ndarray, Rotation, optional
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Unit quaternion in positive real hemisphere.
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Use .from_quaternion to perform a sanity check.
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Defaults to no rotation.
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"""
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Rotation.__init__(self,rotation)
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Crystal.__init__(self,family=family, lattice=lattice,
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a=a,b=b,c=c, alpha=alpha,beta=beta,gamma=gamma, degrees=degrees)
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def __repr__(self):
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"""Represent."""
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return '\n'.join([Crystal.__repr__(self),
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Rotation.__repr__(self)])
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def __copy__(self,rotation=None):
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"""Create deep copy."""
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dup = copy.deepcopy(self)
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if rotation is not None:
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dup.quaternion = Rotation(rotation).quaternion
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return dup
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copy = __copy__
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def __eq__(self,other):
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"""
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Equal to other.
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Parameters
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----------
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other : Orientation
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Orientation to check for equality.
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"""
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matching_type = self.family == other.family and \
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self.lattice == other.lattice and \
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self.parameters == other.parameters
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return np.logical_and(matching_type,super(self.__class__,self.reduced).__eq__(other.reduced))
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def __ne__(self,other):
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"""
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Not equal to other.
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Parameters
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----------
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other : Orientation
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Orientation to check for equality.
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"""
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return np.logical_not(self==other)
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def isclose(self,other,rtol=1e-5,atol=1e-8,equal_nan=True):
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"""
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Report where values are approximately equal to corresponding ones of other Orientation.
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Parameters
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----------
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other : Orientation
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Orientation to compare against.
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rtol : float, optional
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Relative tolerance of equality.
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atol : float, optional
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Absolute tolerance of equality.
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equal_nan : bool, optional
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Consider matching NaN values as equal. Defaults to True.
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Returns
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-------
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mask : numpy.ndarray bool
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Mask indicating where corresponding orientations are close.
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"""
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matching_type = self.family == other.family and \
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self.lattice == other.lattice and \
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self.parameters == other.parameters
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return np.logical_and(matching_type,super(self.__class__,self.reduced).isclose(other.reduced))
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def allclose(self,other,rtol=1e-5,atol=1e-8,equal_nan=True):
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"""
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Test whether all values are approximately equal to corresponding ones of other Orientation.
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Parameters
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----------
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other : Orientation
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Orientation to compare against.
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rtol : float, optional
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Relative tolerance of equality.
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atol : float, optional
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Absolute tolerance of equality.
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equal_nan : bool, optional
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Consider matching NaN values as equal. Defaults to True.
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Returns
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-------
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answer : bool
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Whether all values are close between both orientations.
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"""
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return np.all(self.isclose(other,rtol,atol,equal_nan))
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def __mul__(self,other):
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"""
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Compose this orientation with other.
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Parameters
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----------
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other : Rotation or Orientation
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Object for composition.
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Returns
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-------
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composition : Orientation
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Compound rotation self*other, i.e. first other then self rotation.
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"""
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if isinstance(other,Orientation) or isinstance(other,Rotation):
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return self.copy(rotation=Rotation.__mul__(self,Rotation(other.quaternion)))
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else:
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raise TypeError('use "O@b", i.e. matmul, to apply Orientation "O" to object "b"')
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@staticmethod
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def _split_kwargs(kwargs,target):
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"""
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Separate keyword arguments in 'kwargs' targeted at 'target' from general keyword arguments of Orientation objects.
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Parameters
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----------
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kwargs : dictionary
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Contains all **kwargs.
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target: method
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Function to scan for kwarg signature.
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Returns
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-------
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rot_kwargs: dictionary
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Valid keyword arguments of 'target' function of Rotation class.
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ori_kwargs: dictionary
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Valid keyword arguments of Orientation object.
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"""
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kws = ()
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for t in (target,Orientation.__init__):
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kws += ({key: kwargs[key] for key in set(inspect.signature(t).parameters) & set(kwargs)},)
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invalid_keys = set(kwargs)-(set(kws[0])|set(kws[1]))
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if invalid_keys:
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raise TypeError(f"{inspect.stack()[1][3]}() got an unexpected keyword argument '{invalid_keys.pop()}'")
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return kws
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@classmethod
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@util.extended_docstring(Rotation.from_random,_parameter_doc)
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def from_random(cls,**kwargs):
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kwargs_rot,kwargs_ori = Orientation._split_kwargs(kwargs,Rotation.from_random)
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return cls(rotation=Rotation.from_random(**kwargs_rot),**kwargs_ori)
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@classmethod
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@util.extended_docstring(Rotation.from_quaternion,_parameter_doc)
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def from_quaternion(cls,**kwargs):
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kwargs_rot,kwargs_ori = Orientation._split_kwargs(kwargs,Rotation.from_quaternion)
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return cls(rotation=Rotation.from_quaternion(**kwargs_rot),**kwargs_ori)
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@classmethod
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@util.extended_docstring(Rotation.from_Euler_angles,_parameter_doc)
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def from_Euler_angles(cls,**kwargs):
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kwargs_rot,kwargs_ori = Orientation._split_kwargs(kwargs,Rotation.from_Euler_angles)
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return cls(rotation=Rotation.from_Euler_angles(**kwargs_rot),**kwargs_ori)
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@classmethod
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@util.extended_docstring(Rotation.from_axis_angle,_parameter_doc)
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def from_axis_angle(cls,**kwargs):
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kwargs_rot,kwargs_ori = Orientation._split_kwargs(kwargs,Rotation.from_axis_angle)
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return cls(rotation=Rotation.from_axis_angle(**kwargs_rot),**kwargs_ori)
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@classmethod
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@util.extended_docstring(Rotation.from_basis,_parameter_doc)
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def from_basis(cls,**kwargs):
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kwargs_rot,kwargs_ori = Orientation._split_kwargs(kwargs,Rotation.from_basis)
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return cls(rotation=Rotation.from_basis(**kwargs_rot),**kwargs_ori)
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@classmethod
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@util.extended_docstring(Rotation.from_matrix,_parameter_doc)
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def from_matrix(cls,**kwargs):
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kwargs_rot,kwargs_ori = Orientation._split_kwargs(kwargs,Rotation.from_matrix)
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return cls(rotation=Rotation.from_matrix(**kwargs_rot),**kwargs_ori)
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@classmethod
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@util.extended_docstring(Rotation.from_Rodrigues_vector,_parameter_doc)
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def from_Rodrigues_vector(cls,**kwargs):
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kwargs_rot,kwargs_ori = Orientation._split_kwargs(kwargs,Rotation.from_Rodrigues_vector)
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return cls(rotation=Rotation.from_Rodrigues_vector(**kwargs_rot),**kwargs_ori)
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@classmethod
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@util.extended_docstring(Rotation.from_homochoric,_parameter_doc)
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def from_homochoric(cls,**kwargs):
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kwargs_rot,kwargs_ori = Orientation._split_kwargs(kwargs,Rotation.from_homochoric)
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return cls(rotation=Rotation.from_homochoric(**kwargs_rot),**kwargs_ori)
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@classmethod
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@util.extended_docstring(Rotation.from_cubochoric,_parameter_doc)
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def from_cubochoric(cls,**kwargs):
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kwargs_rot,kwargs_ori = Orientation._split_kwargs(kwargs,Rotation.from_cubochoric)
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return cls(rotation=Rotation.from_cubochoric(**kwargs_rot),**kwargs_ori)
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@classmethod
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@util.extended_docstring(Rotation.from_spherical_component,_parameter_doc)
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def from_spherical_component(cls,**kwargs):
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kwargs_rot,kwargs_ori = Orientation._split_kwargs(kwargs,Rotation.from_spherical_component)
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return cls(rotation=Rotation.from_spherical_component(**kwargs_rot),**kwargs_ori)
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@classmethod
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@util.extended_docstring(Rotation.from_fiber_component,_parameter_doc)
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def from_fiber_component(cls,**kwargs):
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kwargs_rot,kwargs_ori = Orientation._split_kwargs(kwargs,Rotation.from_fiber_component)
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return cls(rotation=Rotation.from_fiber_component(**kwargs_rot),**kwargs_ori)
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@classmethod
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@util.extend_docstring(_parameter_doc)
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def from_directions(cls,uvw,hkl,**kwargs):
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"""
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Initialize orientation object from two crystallographic directions.
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Parameters
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----------
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uvw : list, numpy.ndarray of shape (...,3)
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lattice direction aligned with lab frame x-direction.
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hkl : list, numpy.ndarray of shape (...,3)
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lattice plane normal aligned with lab frame z-direction.
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"""
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o = cls(**kwargs)
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x = o.to_frame(uvw=uvw)
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z = o.to_frame(hkl=hkl)
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om = np.stack([x,np.cross(z,x),z],axis=-2)
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return o.copy(rotation=Rotation.from_matrix(tensor.transpose(om/np.linalg.norm(om,axis=-1,keepdims=True))))
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@property
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def equivalent(self):
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"""
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Orientations that are symmetrically equivalent.
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One dimension (length corresponds to number of symmetrically equivalent orientations)
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is added to the left of the Rotation array.
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"""
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sym_ops = self.symmetry_operations
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o = sym_ops.broadcast_to(sym_ops.shape+self.shape,mode='right')
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return self.copy(rotation=o*Rotation(self.quaternion).broadcast_to(o.shape,mode='left'))
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@property
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def reduced(self):
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"""Select symmetrically equivalent orientation that falls into fundamental zone according to symmetry."""
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eq = self.equivalent
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ok = eq.in_FZ
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ok &= np.cumsum(ok,axis=0) == 1
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loc = np.where(ok)
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sort = 0 if len(loc) == 1 else np.lexsort(loc[:0:-1])
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return eq[ok][sort].reshape(self.shape)
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@property
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def in_FZ(self):
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"""
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Check whether orientation falls into fundamental zone of own symmetry.
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Returns
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-------
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in : numpy.ndarray of bool, quaternion.shape
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Whether Rodrigues-Frank vector falls into fundamental zone.
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Notes
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-----
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Fundamental zones in Rodrigues space are point-symmetric around origin.
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References
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----------
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A. Heinz and P. Neumann, Acta Crystallographica Section A 47:780-789, 1991
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https://doi.org/10.1107/S0108767391006864
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"""
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rho_abs = np.abs(self.as_Rodrigues_vector(compact=True))*(1.-1.e-9)
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with np.errstate(invalid='ignore'):
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# using '*'/prod for 'and'
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if self.family == 'cubic':
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return (np.prod(np.sqrt(2)-1. >= rho_abs,axis=-1) *
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(1. >= np.sum(rho_abs,axis=-1))).astype(bool)
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elif self.family == 'hexagonal':
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return (np.prod(1. >= rho_abs,axis=-1) *
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(2. >= np.sqrt(3)*rho_abs[...,0] + rho_abs[...,1]) *
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(2. >= np.sqrt(3)*rho_abs[...,1] + rho_abs[...,0]) *
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(2. >= np.sqrt(3) + rho_abs[...,2])).astype(bool)
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elif self.family == 'tetragonal':
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return (np.prod(1. >= rho_abs[...,:2],axis=-1) *
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(np.sqrt(2) >= rho_abs[...,0] + rho_abs[...,1]) *
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(np.sqrt(2) >= rho_abs[...,2] + 1.)).astype(bool)
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elif self.family == 'orthorhombic':
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return (np.prod(1. >= rho_abs,axis=-1)).astype(bool)
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elif self.family == 'monoclinic':
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return (1. >= rho_abs[...,1]).astype(bool)
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else:
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return np.all(np.isfinite(rho_abs),axis=-1)
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@property
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def in_disorientation_FZ(self):
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"""
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Check whether orientation falls into fundamental zone of disorientations.
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Returns
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-------
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in : numpy.ndarray of bool, quaternion.shape
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Whether Rodrigues-Frank vector falls into disorientation FZ.
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References
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----------
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A. Heinz and P. Neumann, Acta Crystallographica Section A 47:780-789, 1991
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https://doi.org/10.1107/S0108767391006864
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"""
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rho = self.as_Rodrigues_vector(compact=True)*(1.0-1.0e-9)
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with np.errstate(invalid='ignore'):
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if self.family == 'cubic':
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return ((rho[...,0] >= rho[...,1]) &
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(rho[...,1] >= rho[...,2]) &
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(rho[...,2] >= 0)).astype(bool)
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elif self.family == 'hexagonal':
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return ((rho[...,0] >= rho[...,1]*np.sqrt(3)) &
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(rho[...,1] >= 0) &
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(rho[...,2] >= 0)).astype(bool)
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elif self.family == 'tetragonal':
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return ((rho[...,0] >= rho[...,1]) &
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(rho[...,1] >= 0) &
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(rho[...,2] >= 0)).astype(bool)
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elif self.family == 'orthorhombic':
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return ((rho[...,0] >= 0) &
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(rho[...,1] >= 0) &
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(rho[...,2] >= 0)).astype(bool)
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elif self.family == 'monoclinic':
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return ((rho[...,1] >= 0) &
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(rho[...,2] >= 0)).astype(bool)
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else:
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return np.ones_like(rho[...,0],dtype=bool)
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def disorientation(self,other,return_operators=False):
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"""
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Calculate disorientation between myself and given other orientation.
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Parameters
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----------
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other : Orientation
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Orientation to calculate disorientation for.
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Shape of other blends with shape of own rotation array.
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For example, shapes of (2,3) for own rotations and (3,2) for other's result in (2,3,2) disorientations.
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return_operators : bool, optional
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Return index pair of symmetrically equivalent orientations that result in disorientation axis falling into FZ.
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Defaults to False.
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Returns
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-------
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disorientation : Orientation
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Disorientation between self and other.
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operators : numpy.ndarray int of shape (...,2), conditional
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Index of symmetrically equivalent orientation that rotated vector to the SST.
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Notes
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-----
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Currently requires same crystal family for both orientations.
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For extension to cases with differing symmetry see A. Heinz and P. Neumann 1991 and 10.1107/S0021889808016373.
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Examples
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--------
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Disorientation between two specific orientations of hexagonal symmetry:
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>>> import damask
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>>> a = damask.Orientation.from_Euler_angles(phi=[123,32,21],degrees=True,family='hexagonal')
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>>> b = damask.Orientation.from_Euler_angles(phi=[104,11,87],degrees=True,family='hexagonal')
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>>> a.disorientation(b)
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Crystal family hexagonal
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Quaternion: (real=0.976, imag=<+0.189, +0.018, +0.103>)
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Matrix:
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[[ 0.97831006 0.20710935 0.00389135]
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[-0.19363288 0.90765544 0.37238141]
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[ 0.07359167 -0.36505797 0.92807163]]
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Bunge Eulers / deg: (11.40, 21.86, 0.60)
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Plot a sample from the Mackenzie distribution.
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>>> import matplotlib.pyplot as plt
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>>> import damask
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>>> N = 10000
|
|
>>> a = damask.Orientation.from_random(shape=N,family='cubic')
|
|
>>> b = damask.Orientation.from_random(shape=N,family='cubic')
|
|
>>> d = a.disorientation(b).as_axis_angle(degrees=True,pair=True)[1]
|
|
>>> plt.hist(d,25)
|
|
>>> plt.show()
|
|
|
|
"""
|
|
if self.family != other.family:
|
|
raise NotImplementedError('disorientation between different crystal families')
|
|
|
|
blend = util.shapeblender(self.shape,other.shape)
|
|
s = self.equivalent
|
|
o = other.equivalent
|
|
|
|
s_ = s.reshape((s.shape[0],1)+ self.shape).broadcast_to((s.shape[0],o.shape[0])+blend,mode='right')
|
|
o_ = o.reshape((1,o.shape[0])+other.shape).broadcast_to((s.shape[0],o.shape[0])+blend,mode='right')
|
|
r_ = s_.misorientation(o_)
|
|
_r = ~r_
|
|
|
|
forward = r_.in_FZ & r_.in_disorientation_FZ
|
|
reverse = _r.in_FZ & _r.in_disorientation_FZ
|
|
ok = forward | reverse
|
|
ok &= (np.cumsum(ok.reshape((-1,)+ok.shape[2:]),axis=0) == 1).reshape(ok.shape)
|
|
r = np.where(np.any(forward[...,np.newaxis],axis=(0,1),keepdims=True),
|
|
r_.quaternion,
|
|
_r.quaternion)
|
|
loc = np.where(ok)
|
|
sort = 0 if len(loc) == 2 else np.lexsort(loc[:1:-1])
|
|
quat = r[ok][sort].reshape(blend+(4,))
|
|
|
|
return (
|
|
(self.copy(rotation=quat),
|
|
(np.vstack(loc[:2]).T)[sort].reshape(blend+(2,)))
|
|
if return_operators else
|
|
self.copy(rotation=quat)
|
|
)
|
|
|
|
|
|
def average(self,weights=None,return_cloud=False):
|
|
"""
|
|
Return orientation average over last dimension.
|
|
|
|
Parameters
|
|
----------
|
|
weights : numpy.ndarray, optional
|
|
Relative weights of orientations.
|
|
return_cloud : bool, optional
|
|
Return the set of symmetrically equivalent orientations that was used in averaging.
|
|
Defaults to False.
|
|
|
|
Returns
|
|
-------
|
|
average : Orientation
|
|
Weighted average of original Orientation field.
|
|
cloud : Orientations, conditional
|
|
Set of symmetrically equivalent orientations that were used in averaging.
|
|
|
|
References
|
|
----------
|
|
J.C. Glez and J. Driver, Journal of Applied Crystallography 34:280-288, 2001
|
|
https://doi.org/10.1107/S0021889801003077
|
|
|
|
"""
|
|
eq = self.equivalent
|
|
m = eq.misorientation(self[...,0].reshape((1,)+self.shape[:-1]+(1,))
|
|
.broadcast_to(eq.shape))\
|
|
.as_axis_angle()[...,3]
|
|
r = Rotation(np.squeeze(np.take_along_axis(eq.quaternion,
|
|
np.argmin(m,axis=0)[np.newaxis,...,np.newaxis],
|
|
axis=0),
|
|
axis=0))
|
|
return (
|
|
(self.copy(rotation=Rotation(r).average(weights)),
|
|
self.copy(rotation=Rotation(r)))
|
|
if return_cloud else
|
|
self.copy(rotation=Rotation(r).average(weights))
|
|
)
|
|
|
|
|
|
def to_SST(self,vector,proper=False,return_operators=False):
|
|
"""
|
|
Rotate vector to ensure it falls into (improper or proper) standard stereographic triangle of crystal symmetry.
|
|
|
|
Parameters
|
|
----------
|
|
vector : numpy.ndarray of shape (...,3)
|
|
Lab frame vector to align with crystal frame direction.
|
|
Shape of vector blends with shape of own rotation array.
|
|
For example, a rotation array of shape (3,2) and a (2,4) vector array result in (3,2,4) outputs.
|
|
proper : bool, optional
|
|
Consider only vectors with z >= 0, hence combine two neighboring SSTs.
|
|
Defaults to False.
|
|
return_operators : bool, optional
|
|
Return the symmetrically equivalent orientation that rotated vector to SST.
|
|
Defaults to False.
|
|
|
|
Returns
|
|
-------
|
|
vector_SST : numpy.ndarray of shape (...,3)
|
|
Rotated vector falling into SST.
|
|
operators : numpy.ndarray int of shape (...), conditional
|
|
Index of symmetrically equivalent orientation that rotated vector to SST.
|
|
|
|
"""
|
|
eq = self.equivalent
|
|
blend = util.shapeblender(eq.shape,np.array(vector).shape[:-1])
|
|
poles = eq.broadcast_to(blend,mode='right') @ np.broadcast_to(np.array(vector),blend+(3,))
|
|
ok = self.in_SST(poles,proper=proper)
|
|
ok &= np.cumsum(ok,axis=0) == 1
|
|
loc = np.where(ok)
|
|
sort = 0 if len(loc) == 1 else np.lexsort(loc[:0:-1])
|
|
return (
|
|
(poles[ok][sort].reshape(blend[1:]+(3,)), (np.vstack(loc[:1]).T)[sort].reshape(blend[1:]))
|
|
if return_operators else
|
|
poles[ok][sort].reshape(blend[1:]+(3,))
|
|
)
|
|
|
|
|
|
def in_SST(self,vector,proper=False):
|
|
"""
|
|
Check whether given crystal frame vector falls into standard stereographic triangle of own symmetry.
|
|
|
|
Parameters
|
|
----------
|
|
vector : numpy.ndarray of shape (...,3)
|
|
Vector to check.
|
|
proper : bool, optional
|
|
Consider only vectors with z >= 0, hence combine two neighboring SSTs.
|
|
Defaults to False.
|
|
|
|
Returns
|
|
-------
|
|
in : numpy.ndarray, shape (...)
|
|
Whether vector falls into SST.
|
|
|
|
"""
|
|
if not isinstance(vector,np.ndarray) or vector.shape[-1] != 3:
|
|
raise ValueError('input is not a field of three-dimensional vectors')
|
|
|
|
if self.standard_triangle is None: # direct exit for no symmetry
|
|
return np.ones_like(vector[...,0],bool)
|
|
|
|
if proper:
|
|
components_proper = np.around(np.einsum('...ji,...i',
|
|
np.broadcast_to(self.standard_triangle['proper'], vector.shape+(3,)),
|
|
vector), 12)
|
|
components_improper = np.around(np.einsum('...ji,...i',
|
|
np.broadcast_to(self.standard_triangle['improper'], vector.shape+(3,)),
|
|
vector), 12)
|
|
return np.all(components_proper >= 0.0,axis=-1) \
|
|
| np.all(components_improper >= 0.0,axis=-1)
|
|
else:
|
|
components = np.around(np.einsum('...ji,...i',
|
|
np.broadcast_to(self.standard_triangle['improper'], vector.shape+(3,)),
|
|
np.block([vector[...,:2],np.abs(vector[...,2:3])])), 12)
|
|
|
|
return np.all(components >= 0.0,axis=-1)
|
|
|
|
|
|
def IPF_color(self,vector,in_SST=True,proper=False):
|
|
"""
|
|
Map vector to RGB color within standard stereographic triangle of own symmetry.
|
|
|
|
Parameters
|
|
----------
|
|
vector : numpy.ndarray of shape (...,3)
|
|
Vector to colorize.
|
|
Shape of vector blends with shape of own rotation array.
|
|
For example, a rotation array of shape (3,2) and a (2,4) vector array result in (3,2,4) outputs.
|
|
in_SST : bool, optional
|
|
Consider symmetrically equivalent orientations such that poles are located in SST.
|
|
Defaults to True.
|
|
proper : bool, optional
|
|
Consider only vectors with z >= 0, hence combine two neighboring SSTs (with mirrored colors).
|
|
Defaults to False.
|
|
|
|
Returns
|
|
-------
|
|
rgb : numpy.ndarray of shape (...,3)
|
|
RGB array of IPF colors.
|
|
|
|
Examples
|
|
--------
|
|
Inverse pole figure color of the e_3 direction for a crystal in "Cube" orientation with cubic symmetry:
|
|
|
|
>>> import damask
|
|
>>> o = damask.Orientation(family='cubic')
|
|
>>> o.IPF_color([0,0,1])
|
|
array([1., 0., 0.])
|
|
|
|
"""
|
|
if np.array(vector).shape[-1] != 3:
|
|
raise ValueError('input is not a field of three-dimensional vectors')
|
|
|
|
vector_ = self.to_SST(vector,proper) if in_SST else \
|
|
self @ np.broadcast_to(vector,self.shape+(3,))
|
|
|
|
if self.standard_triangle is None: # direct exit for no symmetry
|
|
return np.zeros_like(vector_)
|
|
|
|
if proper:
|
|
components_proper = np.around(np.einsum('...ji,...i',
|
|
np.broadcast_to(self.standard_triangle['proper'], vector_.shape+(3,)),
|
|
vector_), 12)
|
|
components_improper = np.around(np.einsum('...ji,...i',
|
|
np.broadcast_to(self.standard_triangle['improper'], vector_.shape+(3,)),
|
|
vector_), 12)
|
|
in_SST = np.all(components_proper >= 0.0,axis=-1) \
|
|
| np.all(components_improper >= 0.0,axis=-1)
|
|
components = np.where((in_SST & np.all(components_proper >= 0.0,axis=-1))[...,np.newaxis],
|
|
components_proper,components_improper)
|
|
else:
|
|
components = np.around(np.einsum('...ji,...i',
|
|
np.broadcast_to(self .standard_triangle['improper'], vector_.shape+(3,)),
|
|
np.block([vector_[...,:2],np.abs(vector_[...,2:3])])), 12)
|
|
|
|
in_SST = np.all(components >= 0.0,axis=-1)
|
|
|
|
with np.errstate(invalid='ignore',divide='ignore'):
|
|
rgb = (components/np.linalg.norm(components,axis=-1,keepdims=True))**0.5 # smoothen color ramps
|
|
rgb = np.clip(rgb,0.,1.) # clip intensity
|
|
rgb /= np.max(rgb,axis=-1,keepdims=True) # normalize to (HS)V = 1
|
|
rgb[np.broadcast_to(~in_SST[...,np.newaxis],rgb.shape)] = 0.0
|
|
|
|
return rgb
|
|
|
|
|
|
@property
|
|
def symmetry_operations(self):
|
|
"""Symmetry operations as Rotations."""
|
|
_symmetry_operations = {
|
|
'cubic': [
|
|
[ 1.0, 0.0, 0.0, 0.0 ],
|
|
[ 0.0, 1.0, 0.0, 0.0 ],
|
|
[ 0.0, 0.0, 1.0, 0.0 ],
|
|
[ 0.0, 0.0, 0.0, 1.0 ],
|
|
[ 0.0, 0.0, 0.5*np.sqrt(2), 0.5*np.sqrt(2) ],
|
|
[ 0.0, 0.0, 0.5*np.sqrt(2),-0.5*np.sqrt(2) ],
|
|
[ 0.0, 0.5*np.sqrt(2), 0.0, 0.5*np.sqrt(2) ],
|
|
[ 0.0, 0.5*np.sqrt(2), 0.0, -0.5*np.sqrt(2) ],
|
|
[ 0.0, 0.5*np.sqrt(2),-0.5*np.sqrt(2), 0.0 ],
|
|
[ 0.0, -0.5*np.sqrt(2),-0.5*np.sqrt(2), 0.0 ],
|
|
[ 0.5, 0.5, 0.5, 0.5 ],
|
|
[-0.5, 0.5, 0.5, 0.5 ],
|
|
[-0.5, 0.5, 0.5, -0.5 ],
|
|
[-0.5, 0.5, -0.5, 0.5 ],
|
|
[-0.5, -0.5, 0.5, 0.5 ],
|
|
[-0.5, -0.5, 0.5, -0.5 ],
|
|
[-0.5, -0.5, -0.5, 0.5 ],
|
|
[-0.5, 0.5, -0.5, -0.5 ],
|
|
[-0.5*np.sqrt(2), 0.0, 0.0, 0.5*np.sqrt(2) ],
|
|
[ 0.5*np.sqrt(2), 0.0, 0.0, 0.5*np.sqrt(2) ],
|
|
[-0.5*np.sqrt(2), 0.0, 0.5*np.sqrt(2), 0.0 ],
|
|
[-0.5*np.sqrt(2), 0.0, -0.5*np.sqrt(2), 0.0 ],
|
|
[-0.5*np.sqrt(2), 0.5*np.sqrt(2), 0.0, 0.0 ],
|
|
[-0.5*np.sqrt(2),-0.5*np.sqrt(2), 0.0, 0.0 ],
|
|
],
|
|
'hexagonal': [
|
|
[ 1.0, 0.0, 0.0, 0.0 ],
|
|
[-0.5*np.sqrt(3), 0.0, 0.0, -0.5 ],
|
|
[ 0.5, 0.0, 0.0, 0.5*np.sqrt(3) ],
|
|
[ 0.0, 0.0, 0.0, 1.0 ],
|
|
[-0.5, 0.0, 0.0, 0.5*np.sqrt(3) ],
|
|
[-0.5*np.sqrt(3), 0.0, 0.0, 0.5 ],
|
|
[ 0.0, 1.0, 0.0, 0.0 ],
|
|
[ 0.0, -0.5*np.sqrt(3), 0.5, 0.0 ],
|
|
[ 0.0, 0.5, -0.5*np.sqrt(3), 0.0 ],
|
|
[ 0.0, 0.0, 1.0, 0.0 ],
|
|
[ 0.0, -0.5, -0.5*np.sqrt(3), 0.0 ],
|
|
[ 0.0, 0.5*np.sqrt(3), 0.5, 0.0 ],
|
|
],
|
|
'tetragonal': [
|
|
[ 1.0, 0.0, 0.0, 0.0 ],
|
|
[ 0.0, 1.0, 0.0, 0.0 ],
|
|
[ 0.0, 0.0, 1.0, 0.0 ],
|
|
[ 0.0, 0.0, 0.0, 1.0 ],
|
|
[ 0.0, 0.5*np.sqrt(2), 0.5*np.sqrt(2), 0.0 ],
|
|
[ 0.0, -0.5*np.sqrt(2), 0.5*np.sqrt(2), 0.0 ],
|
|
[ 0.5*np.sqrt(2), 0.0, 0.0, 0.5*np.sqrt(2) ],
|
|
[-0.5*np.sqrt(2), 0.0, 0.0, 0.5*np.sqrt(2) ],
|
|
],
|
|
'orthorhombic': [
|
|
[ 1.0,0.0,0.0,0.0 ],
|
|
[ 0.0,1.0,0.0,0.0 ],
|
|
[ 0.0,0.0,1.0,0.0 ],
|
|
[ 0.0,0.0,0.0,1.0 ],
|
|
],
|
|
'monoclinic': [
|
|
[ 1.0,0.0,0.0,0.0 ],
|
|
[ 0.0,0.0,1.0,0.0 ],
|
|
],
|
|
'triclinic': [
|
|
[ 1.0,0.0,0.0,0.0 ],
|
|
]}
|
|
return Rotation.from_quaternion(_symmetry_operations[self.family],accept_homomorph=True)
|
|
|
|
|
|
####################################################################################################
|
|
# functions that require lattice, not just family
|
|
|
|
def to_pole(self,*,uvw=None,hkl=None,with_symmetry=False):
|
|
"""
|
|
Calculate lab frame vector along lattice direction [uvw] or plane normal (hkl).
|
|
|
|
Parameters
|
|
----------
|
|
uvw|hkl : numpy.ndarray of shape (...,3)
|
|
Miller indices of crystallographic direction or plane normal.
|
|
Shape of vector blends with shape of own rotation array.
|
|
For example, a rotation array of shape (3,2) and a (2,4) vector array result in (3,2,4) outputs.
|
|
with_symmetry : bool, optional
|
|
Calculate all N symmetrically equivalent vectors.
|
|
|
|
Returns
|
|
-------
|
|
vector : numpy.ndarray of shape (...,3) or (...,N,3)
|
|
Lab frame vector (or vectors if with_symmetry) along [uvw] direction or (hkl) plane normal.
|
|
|
|
"""
|
|
v = self.to_frame(uvw=uvw,hkl=hkl)
|
|
blend = util.shapeblender(self.shape,v.shape[:-1])
|
|
if with_symmetry:
|
|
sym_ops = self.symmetry_operations
|
|
shape = v.shape[:-1]+sym_ops.shape
|
|
blend += sym_ops.shape
|
|
v = sym_ops.broadcast_to(shape) \
|
|
@ np.broadcast_to(v.reshape(util.shapeshifter(v.shape,shape+(3,))),shape+(3,))
|
|
return ~(self.broadcast_to(blend)) \
|
|
@ np.broadcast_to(v,blend+(3,))
|
|
|
|
|
|
def Schmid(self,*,N_slip=None,N_twin=None):
|
|
u"""
|
|
Calculate Schmid matrix P = d ⨂ n in the lab frame for selected deformation systems.
|
|
|
|
Parameters
|
|
----------
|
|
N_slip|N_twin : iterable of int
|
|
Number of deformation systems per family of the deformation system.
|
|
Use '*' to select all.
|
|
|
|
Returns
|
|
-------
|
|
P : numpy.ndarray of shape (N,...,3,3)
|
|
Schmid matrix for each of the N deformation systems.
|
|
|
|
Examples
|
|
--------
|
|
Schmid matrix (in lab frame) of first octahedral slip system of a face-centered
|
|
cubic crystal in "Goss" orientation.
|
|
|
|
>>> import damask
|
|
>>> import numpy as np
|
|
>>> np.set_printoptions(3,suppress=True,floatmode='fixed')
|
|
>>> O = damask.Orientation.from_Euler_angles(phi=[0,45,0],degrees=True,lattice='cF')
|
|
>>> O.Schmid(N_slip=[1])
|
|
array([[ 0.000, 0.000, 0.000],
|
|
[ 0.577, -0.000, 0.816],
|
|
[ 0.000, 0.000, 0.000]])
|
|
|
|
"""
|
|
if (N_slip is not None) ^ (N_twin is None):
|
|
raise KeyError('Specify either "N_slip" or "N_twin"')
|
|
|
|
kinematics,active = (self.kinematics('slip'),N_slip) if N_twin is None else \
|
|
(self.kinematics('twin'),N_twin)
|
|
if active == '*': active = [len(a) for a in kinematics['direction']]
|
|
|
|
d = self.to_frame(uvw=np.vstack([kinematics['direction'][i][:n] for i,n in enumerate(active)]))
|
|
p = self.to_frame(hkl=np.vstack([kinematics['plane'][i][:n] for i,n in enumerate(active)]))
|
|
P = np.einsum('...i,...j',d/np.linalg.norm(d,axis=1,keepdims=True),
|
|
p/np.linalg.norm(p,axis=1,keepdims=True))
|
|
|
|
shape = P.shape[0:1]+self.shape+(3,3)
|
|
return ~self.broadcast_to(shape[:-2]) \
|
|
@ np.broadcast_to(P.reshape(util.shapeshifter(P.shape,shape)),shape)
|
|
|
|
|
|
def related(self,model):
|
|
"""
|
|
Orientations derived from the given relationship.
|
|
|
|
One dimension (length according to number of related orientations)
|
|
is added to the left of the Rotation array.
|
|
|
|
"""
|
|
lattice,o = self.relation_operations(model)
|
|
target = Crystal(lattice=lattice)
|
|
o = o.broadcast_to(o.shape+self.shape,mode='right')
|
|
return Orientation(rotation=o*Rotation(self.quaternion).broadcast_to(o.shape,mode='left'),
|
|
lattice=lattice,
|
|
b = self.b if target.ratio['b'] is None else self.a*target.ratio['b'],
|
|
c = self.c if target.ratio['c'] is None else self.a*target.ratio['c'],
|
|
alpha = None if 'alpha' in target.immutable else self.alpha,
|
|
beta = None if 'beta' in target.immutable else self.beta,
|
|
gamma = None if 'gamma' in target.immutable else self.gamma,
|
|
)
|