DAMASK_EICMD/python/damask/_orientation.py

914 lines
36 KiB
Python

import inspect
import copy
import numpy as np
from . import Rotation
from . import Crystal
from . import util
from . import tensor
lattice_symmetries = {
'aP': 'triclinic',
'mP': 'monoclinic',
'mS': 'monoclinic',
'oP': 'orthorhombic',
'oS': 'orthorhombic',
'oI': 'orthorhombic',
'oF': 'orthorhombic',
'tP': 'tetragonal',
'tI': 'tetragonal',
'hP': 'hexagonal',
'cP': 'cubic',
'cI': 'cubic',
'cF': 'cubic',
}
_parameter_doc = \
"""
family : {'triclinic', 'monoclinic', 'orthorhombic', 'tetragonal', 'hexagonal', 'cubic'}, optional.
Name of the crystal family.
Will be infered if 'lattice' is given.
lattice : {'aP', 'mP', 'mS', 'oP', 'oS', 'oI', 'oF', 'tP', 'tI', 'hP', 'cP', 'cI', 'cF'}, optional.
Name of the Bravais lattice in Pearson notation.
a : float, optional
Length of lattice parameter 'a'.
b : float, optional
Length of lattice parameter 'b'.
c : float, optional
Length of lattice parameter 'c'.
alpha : float, optional
Angle between b and c lattice basis.
beta : float, optional
Angle between c and a lattice basis.
gamma : float, optional
Angle between a and b lattice basis.
degrees : bool, optional
Angles are given in degrees. Defaults to False.
"""
class Orientation(Rotation,Crystal):
"""
Representation of crystallographic orientation as combination of rotation and either crystal family or Bravais lattice.
The crystal family is one of:
- triclinic
- monoclinic
- orthorhombic
- tetragonal
- hexagonal
- cubic
and enables symmetry-related operations such as
"equivalent", "reduced", "disorientation", "IPF_color", or "to_SST".
The Bravais lattice is given in the Pearson notation:
- triclinic
- aP : primitive
- monoclinic
- mP : primitive
- mS : base-centered
- orthorhombic
- oP : primitive
- oS : base-centered
- oI : body-centered
- oF : face-centered
- tetragonal
- tP : primitive
- tI : body-centered
- hexagonal
- hP : primitive
- cubic
- cP : primitive
- cI : body-centered
- cF : face-centered
and inherits the corresponding crystal family.
Specifying a Bravais lattice, compared to just the crystal family,
extends the functionality of Orientation objects to include operations such as
"Schmid", "related", or "to_pole" that require a lattice type and its parameters.
Examples
--------
An array of 3 x 5 random orientations reduced to the fundamental zone of tetragonal symmetry:
>>> import damask
>>> o=damask.Orientation.from_random(shape=(3,5),lattice='tetragonal').reduced
"""
@util.extend_docstring(_parameter_doc)
def __init__(self,
rotation = np.array([1.0,0.0,0.0,0.0]), *,
family = None,
lattice = None,
a = None,b = None,c = None,
alpha = None,beta = None,gamma = None,
degrees = False):
"""
New orientation.
Parameters
----------
rotation : list, numpy.ndarray, Rotation, optional
Unit quaternion in positive real hemisphere.
Use .from_quaternion to perform a sanity check.
Defaults to no rotation.
"""
Rotation.__init__(self,rotation)
Crystal.__init__(self,family=family, lattice=lattice,
a=a,b=b,c=c, alpha=alpha,beta=beta,gamma=gamma, degrees=degrees)
def __repr__(self):
"""Represent."""
return '\n'.join(([] if self.lattice is None else [f'Bravais lattice {self.lattice}'])
+ ([f'Crystal family {self.family}'])
+ [super().__repr__()])
def __copy__(self,rotation=None):
"""Create deep copy."""
dup = copy.deepcopy(self)
if rotation is not None:
dup.quaternion = Orientation(rotation,family='cubic').quaternion
return dup
copy = __copy__
def __eq__(self,other):
"""
Equal to other.
Parameters
----------
other : Orientation
Orientation to check for equality.
"""
matching_type = self.family == other.family and \
self.lattice == other.lattice and \
self.parameters == other.parameters
return np.logical_and(matching_type,super(self.__class__,self.reduced).__eq__(other.reduced))
def __ne__(self,other):
"""
Not equal to other.
Parameters
----------
other : Orientation
Orientation to check for equality.
"""
return np.logical_not(self==other)
def isclose(self,other,rtol=1e-5,atol=1e-8,equal_nan=True):
"""
Report where values are approximately equal to corresponding ones of other Orientation.
Parameters
----------
other : Orientation
Orientation to compare against.
rtol : float, optional
Relative tolerance of equality.
atol : float, optional
Absolute tolerance of equality.
equal_nan : bool, optional
Consider matching NaN values as equal. Defaults to True.
Returns
-------
mask : numpy.ndarray bool
Mask indicating where corresponding orientations are close.
"""
matching_type = self.family == other.family and \
self.lattice == other.lattice and \
self.parameters == other.parameters
return np.logical_and(matching_type,super(self.__class__,self.reduced).isclose(other.reduced))
def allclose(self,other,rtol=1e-5,atol=1e-8,equal_nan=True):
"""
Test whether all values are approximately equal to corresponding ones of other Orientation.
Parameters
----------
other : Orientation
Orientation to compare against.
rtol : float, optional
Relative tolerance of equality.
atol : float, optional
Absolute tolerance of equality.
equal_nan : bool, optional
Consider matching NaN values as equal. Defaults to True.
Returns
-------
answer : bool
Whether all values are close between both orientations.
"""
return np.all(self.isclose(other,rtol,atol,equal_nan))
def __mul__(self,other):
"""
Compose this orientation with other.
Parameters
----------
other : Rotation or Orientation
Object for composition.
Returns
-------
composition : Orientation
Compound rotation self*other, i.e. first other then self rotation.
"""
if isinstance(other,Orientation) or isinstance(other,Rotation):
return self.copy(rotation=Rotation.__mul__(self,Rotation(other.quaternion)))
else:
raise TypeError('use "O@b", i.e. matmul, to apply Orientation "O" to object "b"')
@staticmethod
def _split_kwargs(kwargs,target):
"""
Separate keyword arguments in 'kwargs' targeted at 'target' from general keyword arguments of Orientation objects.
Parameters
----------
kwargs : dictionary
Contains all **kwargs.
target: method
Function to scan for kwarg signature.
Returns
-------
rot_kwargs: dictionary
Valid keyword arguments of 'target' function of Rotation class.
ori_kwargs: dictionary
Valid keyword arguments of Orientation object.
"""
kws = ()
for t in (target,Orientation.__init__):
kws += ({key: kwargs[key] for key in set(inspect.signature(t).parameters) & set(kwargs)},)
invalid_keys = set(kwargs)-(set(kws[0])|set(kws[1]))
if invalid_keys:
raise TypeError(f"{inspect.stack()[1][3]}() got an unexpected keyword argument '{invalid_keys.pop()}'")
return kws
@classmethod
@util.extended_docstring(Rotation.from_random,_parameter_doc)
def from_random(cls,**kwargs):
kwargs_rot,kwargs_ori = Orientation._split_kwargs(kwargs,Rotation.from_random)
return cls(rotation=Rotation.from_random(**kwargs_rot),**kwargs_ori)
@classmethod
@util.extended_docstring(Rotation.from_quaternion,_parameter_doc)
def from_quaternion(cls,**kwargs):
kwargs_rot,kwargs_ori = Orientation._split_kwargs(kwargs,Rotation.from_quaternion)
return cls(rotation=Rotation.from_quaternion(**kwargs_rot),**kwargs_ori)
@classmethod
@util.extended_docstring(Rotation.from_Euler_angles,_parameter_doc)
def from_Euler_angles(cls,**kwargs):
kwargs_rot,kwargs_ori = Orientation._split_kwargs(kwargs,Rotation.from_Euler_angles)
return cls(rotation=Rotation.from_Euler_angles(**kwargs_rot),**kwargs_ori)
@classmethod
@util.extended_docstring(Rotation.from_axis_angle,_parameter_doc)
def from_axis_angle(cls,**kwargs):
kwargs_rot,kwargs_ori = Orientation._split_kwargs(kwargs,Rotation.from_axis_angle)
return cls(rotation=Rotation.from_axis_angle(**kwargs_rot),**kwargs_ori)
@classmethod
@util.extended_docstring(Rotation.from_basis,_parameter_doc)
def from_basis(cls,**kwargs):
kwargs_rot,kwargs_ori = Orientation._split_kwargs(kwargs,Rotation.from_basis)
return cls(rotation=Rotation.from_basis(**kwargs_rot),**kwargs_ori)
@classmethod
@util.extended_docstring(Rotation.from_matrix,_parameter_doc)
def from_matrix(cls,**kwargs):
kwargs_rot,kwargs_ori = Orientation._split_kwargs(kwargs,Rotation.from_matrix)
return cls(rotation=Rotation.from_matrix(**kwargs_rot),**kwargs_ori)
@classmethod
@util.extended_docstring(Rotation.from_Rodrigues_vector,_parameter_doc)
def from_Rodrigues_vector(cls,**kwargs):
kwargs_rot,kwargs_ori = Orientation._split_kwargs(kwargs,Rotation.from_Rodrigues_vector)
return cls(rotation=Rotation.from_Rodrigues_vector(**kwargs_rot),**kwargs_ori)
@classmethod
@util.extended_docstring(Rotation.from_homochoric,_parameter_doc)
def from_homochoric(cls,**kwargs):
kwargs_rot,kwargs_ori = Orientation._split_kwargs(kwargs,Rotation.from_homochoric)
return cls(rotation=Rotation.from_homochoric(**kwargs_rot),**kwargs_ori)
@classmethod
@util.extended_docstring(Rotation.from_cubochoric,_parameter_doc)
def from_cubochoric(cls,**kwargs):
kwargs_rot,kwargs_ori = Orientation._split_kwargs(kwargs,Rotation.from_cubochoric)
return cls(rotation=Rotation.from_cubochoric(**kwargs_rot),**kwargs_ori)
@classmethod
@util.extended_docstring(Rotation.from_spherical_component,_parameter_doc)
def from_spherical_component(cls,**kwargs):
kwargs_rot,kwargs_ori = Orientation._split_kwargs(kwargs,Rotation.from_spherical_component)
return cls(rotation=Rotation.from_spherical_component(**kwargs_rot),**kwargs_ori)
@classmethod
@util.extended_docstring(Rotation.from_fiber_component,_parameter_doc)
def from_fiber_component(cls,**kwargs):
kwargs_rot,kwargs_ori = Orientation._split_kwargs(kwargs,Rotation.from_fiber_component)
return cls(rotation=Rotation.from_fiber_component(**kwargs_rot),**kwargs_ori)
@classmethod
@util.extend_docstring(_parameter_doc)
def from_directions(cls,uvw,hkl,**kwargs):
"""
Initialize orientation object from two crystallographic directions.
Parameters
----------
uvw : list, numpy.ndarray of shape (...,3)
lattice direction aligned with lab frame x-direction.
hkl : list, numpy.ndarray of shape (...,3)
lattice plane normal aligned with lab frame z-direction.
"""
o = cls(**kwargs)
x = o.to_frame(uvw=uvw)
z = o.to_frame(hkl=hkl)
om = np.stack([x,np.cross(z,x),z],axis=-2)
return o.copy(rotation=Rotation.from_matrix(tensor.transpose(om/np.linalg.norm(om,axis=-1,keepdims=True))))
@property
def equivalent(self):
"""
Orientations that are symmetrically equivalent.
One dimension (length corresponds to number of symmetrically equivalent orientations)
is added to the left of the Rotation array.
"""
sym_ops = self.symmetry_operations
o = sym_ops.broadcast_to(sym_ops.shape+self.shape,mode='right')
return self.copy(rotation=o*Rotation(self.quaternion).broadcast_to(o.shape,mode='left'))
@property
def reduced(self):
"""Select symmetrically equivalent orientation that falls into fundamental zone according to symmetry."""
eq = self.equivalent
ok = eq.in_FZ
ok &= np.cumsum(ok,axis=0) == 1
loc = np.where(ok)
sort = 0 if len(loc) == 1 else np.lexsort(loc[:0:-1])
return eq[ok][sort].reshape(self.shape)
@property
def in_FZ(self):
"""
Check whether orientation falls into fundamental zone of own symmetry.
Returns
-------
in : numpy.ndarray of quaternion.shape
Boolean array indicating whether Rodrigues-Frank vector falls into fundamental zone.
Notes
-----
Fundamental zones in Rodrigues space are point-symmetric around origin.
References
----------
A. Heinz and P. Neumann, Acta Crystallographica Section A 47:780-789, 1991
https://doi.org/10.1107/S0108767391006864
"""
rho_abs = np.abs(self.as_Rodrigues_vector(compact=True))*(1.-1.e-9)
with np.errstate(invalid='ignore'):
# using '*'/prod for 'and'
if self.family == 'cubic':
return (np.prod(np.sqrt(2)-1. >= rho_abs,axis=-1) *
(1. >= np.sum(rho_abs,axis=-1))).astype(bool)
elif self.family == 'hexagonal':
return (np.prod(1. >= rho_abs,axis=-1) *
(2. >= np.sqrt(3)*rho_abs[...,0] + rho_abs[...,1]) *
(2. >= np.sqrt(3)*rho_abs[...,1] + rho_abs[...,0]) *
(2. >= np.sqrt(3) + rho_abs[...,2])).astype(bool)
elif self.family == 'tetragonal':
return (np.prod(1. >= rho_abs[...,:2],axis=-1) *
(np.sqrt(2) >= rho_abs[...,0] + rho_abs[...,1]) *
(np.sqrt(2) >= rho_abs[...,2] + 1.)).astype(bool)
elif self.family == 'orthorhombic':
return (np.prod(1. >= rho_abs,axis=-1)).astype(bool)
elif self.family == 'monoclinic':
return (1. >= rho_abs[...,1]).astype(bool)
else:
return np.all(np.isfinite(rho_abs),axis=-1)
@property
def in_disorientation_FZ(self):
"""
Check whether orientation falls into fundamental zone of disorientations.
Returns
-------
in : numpy.ndarray of quaternion.shape
Boolean array indicating whether Rodrigues-Frank vector falls into disorientation FZ.
References
----------
A. Heinz and P. Neumann, Acta Crystallographica Section A 47:780-789, 1991
https://doi.org/10.1107/S0108767391006864
"""
rho = self.as_Rodrigues_vector(compact=True)*(1.0-1.0e-9)
with np.errstate(invalid='ignore'):
if self.family == 'cubic':
return ((rho[...,0] >= rho[...,1]) &
(rho[...,1] >= rho[...,2]) &
(rho[...,2] >= 0)).astype(bool)
elif self.family == 'hexagonal':
return ((rho[...,0] >= rho[...,1]*np.sqrt(3)) &
(rho[...,1] >= 0) &
(rho[...,2] >= 0)).astype(bool)
elif self.family == 'tetragonal':
return ((rho[...,0] >= rho[...,1]) &
(rho[...,1] >= 0) &
(rho[...,2] >= 0)).astype(bool)
elif self.family == 'orthorhombic':
return ((rho[...,0] >= 0) &
(rho[...,1] >= 0) &
(rho[...,2] >= 0)).astype(bool)
elif self.family == 'monoclinic':
return ((rho[...,1] >= 0) &
(rho[...,2] >= 0)).astype(bool)
else:
return np.ones_like(rho[...,0],dtype=bool)
def disorientation(self,other,return_operators=False):
"""
Calculate disorientation between myself and given other orientation.
Parameters
----------
other : Orientation
Orientation to calculate disorientation for.
Shape of other blends with shape of own rotation array.
For example, shapes of (2,3) for own rotations and (3,2) for other's result in (2,3,2) disorientations.
return_operators : bool, optional
Return index pair of symmetrically equivalent orientations that result in disorientation axis falling into FZ.
Defaults to False.
Returns
-------
disorientation : Orientation
Disorientation between self and other.
operators : numpy.ndarray int of shape (...,2), conditional
Index of symmetrically equivalent orientation that rotated vector to the SST.
Notes
-----
Currently requires same crystal family for both orientations.
For extension to cases with differing symmetry see A. Heinz and P. Neumann 1991 and 10.1107/S0021889808016373.
Examples
--------
Disorientation between two specific orientations of hexagonal symmetry:
>>> import damask
>>> a = damask.Orientation.from_Eulers(phi=[123,32,21],degrees=True,lattice='hexagonal')
>>> b = damask.Orientation.from_Eulers(phi=[104,11,87],degrees=True,lattice='hexagonal')
>>> a.disorientation(b)
Crystal family hexagonal
Quaternion: (real=0.976, imag=<+0.189, +0.018, +0.103>)
Matrix:
[[ 0.97831006 0.20710935 0.00389135]
[-0.19363288 0.90765544 0.37238141]
[ 0.07359167 -0.36505797 0.92807163]]
Bunge Eulers / deg: (11.40, 21.86, 0.60)
"""
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 other 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 of shape (...)
Boolean array indicating 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.
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(lattice='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.
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)
if with_symmetry:
sym_ops = self.symmetry_operations
v = sym_ops.broadcast_to(sym_ops.shape+v.shape[:-1],mode='right') \
@ np.broadcast_to(v,sym_ops.shape+v.shape)
return ~(self if self.shape+v.shape[:-1] == () else self.broadcast_to(self.shape+v.shape[:-1],mode='right')) \
@ np.broadcast_to(v,self.shape+v.shape)
def Schmid(self,mode):
u"""
Calculate Schmid matrix P = d ⨂ n in the lab frame for given lattice shear kinematics.
Parameters
----------
mode : {'slip', 'twin'}
Deformation mode.
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 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')
>>> damask.Orientation.from_Eulers(phi=[0,45,0],degrees=True,lattice='cF').Schmid('slip')[0]
array([[ 0.000, 0.000, 0.000],
[ 0.577, -0.000, 0.816],
[ 0.000, 0.000, 0.000]])
"""
d = self.to_frame(uvw=self.kinematics(mode)['direction'])
p = self.to_frame(hkl=self.kinematics(mode)['plane'])
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,
)