DAMASK_EICMD/python/damask/_orientation.py

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import numpy as np
from . import Rotation
from . import util
from . import mechanics
_parameter_doc = \
"""lattice : str
Either a crystal family out of [triclinic, monoclinic, orthorhombic, tetragonal, hexagonal, cubic]
or a Bravais lattice out of [aP, mP, mS, oP, oS, oI, oF, tP, tI, hP, cP, cI, cF].
When specifying a Bravais lattice, additional lattice parameters might be required:
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):
"""
Representation of crystallographic orientation as combination of rotation and either crystal family or Bravais lattice.
The crystal family is one of Orientation.crystal_families:
- 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 one of Orientation.lattice_symmetries:
- aP : triclinic primitive
- mP : monoclinic primitive
- mS : ... base-centered
- oP : orthorhombic primitive
- oS : ... base-centered
- oI : ... body-centered
- oF : ... face-centered
- tP : tetragonal primitive
- tI : ... body-centered
- hP : hexagonal primitive
- cP : cubic 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:
>>> damask.Orientation.from_random(shape=(3,5),lattice='tetragonal').reduced
"""
crystal_families = ['triclinic',
'monoclinic',
'orthorhombic',
'tetragonal',
'hexagonal',
'cubic']
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',
}
@util.extend_docstring(_parameter_doc)
def __init__(self,
rotation = None,
lattice = None,
a = None,b = None,c = None,
alpha = None,beta = None,gamma = None,
degrees = False,
**kwargs):
"""
Initialize orientation object.
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.
"""
from damask.lattice import kinematics
Rotation.__init__(self) if rotation is None else Rotation.__init__(self,rotation=rotation)
if ( lattice is not None
and lattice not in self.lattice_symmetries
and lattice not in self.crystal_families):
raise KeyError(f'Lattice "{lattice}" is unknown')
self.family = None
self.lattice = None
self.a = None
self.b = None
self.c = None
self.alpha = None
self.beta = None
self.gamma = None
self.kinematics = None
if lattice in self.lattice_symmetries:
self.family = self.lattice_symmetries[lattice]
self.lattice = lattice
self.a = 1 if a is None else a
self.b = b
self.c = c
self.alpha = (np.radians(alpha) if degrees else alpha) if alpha is not None else None
self.beta = (np.radians(beta) if degrees else beta) if beta is not None else None
self.gamma = (np.radians(gamma) if degrees else gamma) if gamma is not None else None
self.a = float(self.a) if self.a is not None else \
(self.b / self.ratio['b'] if self.b is not None and self.ratio['b'] is not None else
self.c / self.ratio['c'] if self.c is not None and self.ratio['c'] is not None else None)
self.b = float(self.b) if self.b is not None else \
(self.a * self.ratio['b'] if self.a is not None and self.ratio['b'] is not None else
self.c / self.ratio['c'] * self.ratio['b']
if self.c is not None and self.ratio['b'] is not None and self.ratio['c'] is not None else None)
self.c = float(self.c) if self.c is not None else \
(self.a * self.ratio['c'] if self.a is not None and self.ratio['c'] is not None else
self.b / self.ratio['b'] * self.ratio['c']
if self.c is not None and self.ratio['b'] is not None and self.ratio['c'] is not None else None)
self.alpha = self.alpha if self.alpha is not None else self.immutable['alpha'] if 'alpha' in self.immutable else None
self.beta = self.beta if self.beta is not None else self.immutable['beta'] if 'beta' in self.immutable else None
self.gamma = self.gamma if self.gamma is not None else self.immutable['gamma'] if 'gamma' in self.immutable else None
if \
(self.a is None) \
or (self.b is None or ('b' in self.immutable and self.b != self.immutable['b'] * self.a)) \
or (self.c is None or ('c' in self.immutable and self.c != self.immutable['c'] * self.b)) \
or (self.alpha is None or ('alpha' in self.immutable and self.alpha != self.immutable['alpha'])) \
or (self.beta is None or ( 'beta' in self.immutable and self.beta != self.immutable['beta'])) \
or (self.gamma is None or ('gamma' in self.immutable and self.gamma != self.immutable['gamma'])):
raise ValueError (f'Incompatible parameters {self.parameters} for crystal family {self.family}')
if np.any(np.array([self.alpha,self.beta,self.gamma]) <= 0):
raise ValueError ('Lattice angles must be positive')
if np.any([np.roll([self.alpha,self.beta,self.gamma],r)[0]
> np.sum(np.roll([self.alpha,self.beta,self.gamma],r)[1:]) for r in range(3)]):
raise ValueError ('Each lattice angle must be less than sum of others')
if self.lattice in kinematics:
master = kinematics[self.lattice]
self.kinematics = {}
for m in master:
self.kinematics[m] = {'direction':master[m][:,0:3],'plane':master[m][:,3:6]} \
if master[m].shape[-1] == 6 else \
{'direction':self.Bravais_to_Miller(uvtw=master[m][:,0:4]),
'plane': self.Bravais_to_Miller(hkil=master[m][:,4:8])}
elif lattice in self.crystal_families:
self.family = lattice
def __repr__(self):
"""Represent."""
return '\n'.join(([] if self.lattice is None else [f'Bravais lattice {self.lattice}'])
+ ([] if self.family is None else [f'Crystal family {self.family}'])
+ [super().__repr__()])
def __copy__(self,**kwargs):
"""Copy."""
return self.__class__(rotation=kwargs['rotation'] if 'rotation' in kwargs else self.quaternion,
lattice =kwargs['lattice'] if 'lattice' in kwargs else self.lattice
if self.lattice is not None else self.family,
a =kwargs['a'] if 'a' in kwargs else self.a,
b =kwargs['b'] if 'b' in kwargs else self.b,
c =kwargs['c'] if 'c' in kwargs else self.c,
alpha =kwargs['alpha'] if 'alpha' in kwargs else self.alpha,
beta =kwargs['beta'] if 'beta' in kwargs else self.beta,
gamma =kwargs['gamma'] if 'gamma' in kwargs else self.gamma,
degrees =kwargs['degrees'] if 'degrees' in kwargs else None,
)
copy = __copy__
def __eq__(self,other):
"""
Equal to other.
Parameters
----------
other : Orientation
Orientation to check for equality.
"""
return super().__eq__(other) \
and self.family == other.family \
and self.lattice == other.lattice \
and self.parameters == other.parameters
def __matmul__(self,other):
"""
Rotation of vector, second or fourth order tensor, or rotation object.
Parameters
----------
other : numpy.ndarray, Rotation, or Orientation
Vector, second or fourth order tensor, or rotation object that is rotated.
Returns
-------
other_rot : numpy.ndarray or Rotation
Rotated vector, second or fourth order tensor, or rotation object.
"""
return self.copy(rotation=Rotation.__matmul__(self,Rotation(other.quaternion))) \
if isinstance(other,self.__class__) else \
Rotation.__matmul__(self,other)
@classmethod
@util.extended_docstring(Rotation.from_random,_parameter_doc)
def from_random(cls,**kwargs):
return cls(rotation=Rotation.from_random(**kwargs),**kwargs)
@classmethod
@util.extended_docstring(Rotation.from_quaternion,_parameter_doc)
def from_quaternion(cls,**kwargs):
return cls(rotation=Rotation.from_quaternion(**kwargs),**kwargs)
@classmethod
@util.extended_docstring(Rotation.from_Eulers,_parameter_doc)
def from_Eulers(cls,**kwargs):
return cls(rotation=Rotation.from_Eulers(**kwargs),**kwargs)
@classmethod
@util.extended_docstring(Rotation.from_axis_angle,_parameter_doc)
def from_axis_angle(cls,**kwargs):
return cls(rotation=Rotation.from_axis_angle(**kwargs),**kwargs)
@classmethod
@util.extended_docstring(Rotation.from_basis,_parameter_doc)
def from_basis(cls,**kwargs):
return cls(rotation=Rotation.from_basis(**kwargs),**kwargs)
@classmethod
@util.extended_docstring(Rotation.from_matrix,_parameter_doc)
def from_matrix(cls,**kwargs):
return cls(rotation=Rotation.from_matrix(**kwargs),**kwargs)
@classmethod
@util.extended_docstring(Rotation.from_Rodrigues,_parameter_doc)
def from_Rodrigues(cls,**kwargs):
return cls(rotation=Rotation.from_Rodrigues(**kwargs),**kwargs)
@classmethod
@util.extended_docstring(Rotation.from_homochoric,_parameter_doc)
def from_homochoric(cls,**kwargs):
return cls(rotation=Rotation.from_homochoric(**kwargs),**kwargs)
@classmethod
@util.extended_docstring(Rotation.from_cubochoric,_parameter_doc)
def from_cubochoric(cls,**kwargs):
return cls(rotation=Rotation.from_cubochoric(**kwargs),**kwargs)
@classmethod
@util.extended_docstring(Rotation.from_spherical_component,_parameter_doc)
def from_spherical_component(cls,**kwargs):
return cls(rotation=Rotation.from_spherical_component(**kwargs),**kwargs)
@classmethod
@util.extended_docstring(Rotation.from_fiber_component,_parameter_doc)
def from_fiber_component(cls,**kwargs):
return cls(rotation=Rotation.from_fiber_component(**kwargs),**kwargs)
@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(mechanics.transpose(om/np.linalg.norm(om,axis=-1,keepdims=True))))
@property
def symmetry_operations(self):
"""Symmetry operations as Rotations."""
if self.family == 'cubic':
sym_quats = [
[ 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 ],
]
elif self.family == 'hexagonal':
sym_quats = [
[ 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 ],
]
elif self.family == 'tetragonal':
sym_quats = [
[ 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) ],
]
elif self.family == 'orthorhombic':
sym_quats = [
[ 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 ],
]
elif self.family == 'monoclinic':
sym_quats = [
[ 1.0,0.0,0.0,0.0 ],
[ 0.0,0.0,1.0,0.0 ],
]
elif self.family == 'triclinic':
sym_quats = [
[ 1.0,0.0,0.0,0.0 ],
]
else:
raise KeyError(f'Crystal family "{self.family}" is unknown')
return Rotation.from_quaternion(sym_quats,accept_homomorph=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.
"""
if self.family is None:
raise ValueError('Missing crystal symmetry')
o = self.symmetry_operations.broadcast_to(self.symmetry_operations.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."""
if self.family is None:
raise ValueError('Missing crystal 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
"""
if self.family is None:
raise ValueError('Missing crystal symmetry')
rho_abs = np.abs(self.as_Rodrigues(vector=True))
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(np.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(np.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(np.bool)
elif self.family == 'orthorhombic':
return (np.prod(1. >= rho_abs,axis=-1)).astype(np.bool)
elif self.family == 'monoclinic':
return (1. >= rho_abs[...,1]).astype(np.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
"""
if self.family is None:
raise ValueError('Missing crystal symmetry')
rho = self.as_Rodrigues(vector=True)
with np.errstate(invalid='ignore'):
if self.family == 'cubic':
return ((rho[...,0] >= rho[...,1]) &
(rho[...,1] >= rho[...,2]) &
(rho[...,2] >= 0)).astype(np.bool)
elif self.family == 'hexagonal':
return ((rho[...,0] >= rho[...,1]*np.sqrt(3)) &
(rho[...,1] >= 0) &
(rho[...,2] >= 0)).astype(np.bool)
elif self.family == 'tetragonal':
return ((rho[...,0] >= rho[...,1]) &
(rho[...,1] >= 0) &
(rho[...,2] >= 0)).astype(np.bool)
elif self.family == 'orthorhombic':
return ((rho[...,0] >= 0) &
(rho[...,1] >= 0) &
(rho[...,2] >= 0)).astype(np.bool)
elif self.family == 'monoclinic':
return ((rho[...,1] >= 0) &
(rho[...,2] >= 0)).astype(np.bool)
else:
return np.ones_like(rho[...,0],dtype=bool)
def relation_operations(self,model,return_lattice=False):
"""
Crystallographic orientation relationships for phase transformations.
Parameters
----------
model : str
Name of orientation relationship.
return_lattice : bool, optional
Return the target lattice in addition.
Returns
-------
operations : Rotations
Rotations characterizing the orientation relationship.
References
----------
S. Morito et al., Journal of Alloys and Compounds 577:s587-s592, 2013
https://doi.org/10.1016/j.jallcom.2012.02.004
K. Kitahara et al., Acta Materialia 54(5):1279-1288, 2006
https://doi.org/10.1016/j.actamat.2005.11.001
Y. He et al., Journal of Applied Crystallography 39:72-81, 2006
https://doi.org/10.1107/S0021889805038276
H. Kitahara et al., Materials Characterization 54(4-5):378-386, 2005
https://doi.org/10.1016/j.matchar.2004.12.015
Y. He et al., Acta Materialia 53(4):1179-1190, 2005
https://doi.org/10.1016/j.actamat.2004.11.021
"""
from damask.lattice import relations
if model not in relations:
raise KeyError(f'Orientation relationship "{model}" is unknown')
r = relations[model]
if self.lattice not in r:
raise KeyError(f'Relationship "{model}" not supported for lattice "{self.lattice}"')
sl = self.lattice
ol = (set(r)-{sl}).pop()
m = r[sl]
o = r[ol]
p_,_p = np.zeros(m.shape[:-1]+(3,)),np.zeros(o.shape[:-1]+(3,))
p_[...,0,:] = m[...,0,:] if m.shape[-1] == 3 else self.Bravais_to_Miller(uvtw=m[...,0,0:4])
p_[...,1,:] = m[...,1,:] if m.shape[-1] == 3 else self.Bravais_to_Miller(hkil=m[...,1,0:4])
_p[...,0,:] = o[...,0,:] if o.shape[-1] == 3 else self.Bravais_to_Miller(uvtw=o[...,0,0:4])
_p[...,1,:] = o[...,1,:] if o.shape[-1] == 3 else self.Bravais_to_Miller(hkil=o[...,1,0:4])
return (Rotation.from_parallel(p_,_p),ol) \
if return_lattice else \
Rotation.from_parallel(p_,_p)
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.
"""
o,lattice = self.relation_operations(model,return_lattice=True)
target = Orientation(lattice=lattice)
o = o.broadcast_to(o.shape+self.shape,mode='right')
return self.copy(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,
)
@property
def parameters(self):
"""Return lattice parameters a, b, c, alpha, beta, gamma."""
return (self.a,self.b,self.c,self.alpha,self.beta,self.gamma)
@property
def immutable(self):
"""Return immutable parameters of own lattice."""
if self.family == 'triclinic':
return {}
if self.family == 'monoclinic':
return {
'alpha': np.pi/2.,
'gamma': np.pi/2.,
}
if self.family == 'orthorhombic':
return {
'alpha': np.pi/2.,
'beta': np.pi/2.,
'gamma': np.pi/2.,
}
if self.family == 'tetragonal':
return {
'b': 1.0,
'alpha': np.pi/2.,
'beta': np.pi/2.,
'gamma': np.pi/2.,
}
if self.family == 'hexagonal':
return {
'b': 1.0,
'alpha': np.pi/2.,
'beta': np.pi/2.,
'gamma': 2.*np.pi/3.,
}
if self.family == 'cubic':
return {
'b': 1.0,
'c': 1.0,
'alpha': np.pi/2.,
'beta': np.pi/2.,
'gamma': np.pi/2.,
}
else:
raise KeyError(f'Crystal family "{self.family}" is unknown')
@property
def ratio(self):
"""Return axes ratios of own lattice."""
_ratio = { 'hexagonal': {'c': np.sqrt(8./3.)}}
return dict(b = self.immutable['b']
if 'b' in self.immutable else
_ratio[self.family]['b'] if self.family in _ratio and 'b' in _ratio[self.family] else None,
c = self.immutable['c']
if 'c' in self.immutable else
_ratio[self.family]['c'] if self.family in _ratio and 'c' in _ratio[self.family] else None,
)
@property
def basis_real(self):
"""
Calculate orthogonal real space crystal basis.
References
----------
C.T. Young and J.L. Lytton, J. Appl. Phys. 43:14081417, 1972
"Computer Generation and Identification of Kikuchi Projections"
https://doi.org/10.1063/1.1661333
"""
if None in self.parameters:
raise KeyError('Missing crystal lattice parameters')
return np.array([
[1,0,0],
[np.cos(self.gamma),np.sin(self.gamma),0],
[np.cos(self.beta),
(np.cos(self.alpha)-np.cos(self.beta)*np.cos(self.gamma)) /np.sin(self.gamma),
np.sqrt(1 - np.cos(self.alpha)**2 - np.cos(self.beta)**2 - np.cos(self.gamma)**2
+ 2 * np.cos(self.alpha) * np.cos(self.beta) * np.cos(self.gamma))/np.sin(self.gamma)],
],dtype=float).T \
* np.array([self.a,self.b,self.c])
@property
def basis_reciprocal(self):
"""Calculate reciprocal (dual) crystal basis."""
return np.linalg.inv(self.basis_real.T)
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.
References
----------
Bases are computed from
>>> basis = {
... 'cubic' : np.linalg.inv(np.array([[0.,0.,1.], # direction of red
... [1.,0.,1.]/np.sqrt(2.), # green
... [1.,1.,1.]/np.sqrt(3.)]).T), # blue
... 'hexagonal' : np.linalg.inv(np.array([[0.,0.,1.], # direction of red
... [1.,0.,0.], # green
... [np.sqrt(3.),1.,0.]/np.sqrt(4.)]).T), # blue
... 'tetragonal' : np.linalg.inv(np.array([[0.,0.,1.], # direction of red
... [1.,0.,0.], # green
... [1.,1.,0.]/np.sqrt(2.)]).T), # blue
... 'orthorhombic': np.linalg.inv(np.array([[0.,0.,1.], # direction of red
... [1.,0.,0.], # green
... [0.,1.,0.]]).T), # blue
... }
"""
if not isinstance(vector,np.ndarray) or vector.shape[-1] != 3:
raise ValueError('Input is not a field of three-dimensional vectors.')
if self.family == 'cubic':
basis = {'improper':np.array([ [-1. , 0. , 1. ],
[ np.sqrt(2.) , -np.sqrt(2.) , 0. ],
[ 0. , np.sqrt(3.) , 0. ] ]),
'proper':np.array([ [ 0. , -1. , 1. ],
[-np.sqrt(2.) , np.sqrt(2.) , 0. ],
[ np.sqrt(3.) , 0. , 0. ] ]),
}
elif self.family == 'hexagonal':
basis = {'improper':np.array([ [ 0. , 0. , 1. ],
[ 1. , -np.sqrt(3.) , 0. ],
[ 0. , 2. , 0. ] ]),
'proper':np.array([ [ 0. , 0. , 1. ],
[-1. , np.sqrt(3.) , 0. ],
[ np.sqrt(3.) , -1. , 0. ] ]),
}
elif self.family == 'tetragonal':
basis = {'improper':np.array([ [ 0. , 0. , 1. ],
[ 1. , -1. , 0. ],
[ 0. , np.sqrt(2.) , 0. ] ]),
'proper':np.array([ [ 0. , 0. , 1. ],
[-1. , 1. , 0. ],
[ np.sqrt(2.) , 0. , 0. ] ]),
}
elif self.family == 'orthorhombic':
basis = {'improper':np.array([ [ 0., 0., 1.],
[ 1., 0., 0.],
[ 0., 1., 0.] ]),
'proper':np.array([ [ 0., 0., 1.],
[-1., 0., 0.],
[ 0., 1., 0.] ]),
}
else: # direct exit for unspecified symmetry
return np.ones_like(vector[...,0],bool)
if proper:
components_proper = np.around(np.einsum('...ji,...i',
np.broadcast_to(basis['proper'], vector.shape+(3,)),
vector), 12)
components_improper = np.around(np.einsum('...ji,...i',
np.broadcast_to(basis['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(basis['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,proper=False):
"""
Map vector to RGB color within standard stereographic triangle of own symmetry.
Parameters
----------
vector : numpy.ndarray of shape (...,3)
Vector to colorize.
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:
>>> o = damask.Orientation(lattice='cubic')
>>> o.IPF_color(o.to_SST([0,0,1]))
array([1., 0., 0.])
References
----------
Bases are computed from
>>> basis = {
... 'cubic' : np.linalg.inv(np.array([[0.,0.,1.], # direction of red
... [1.,0.,1.]/np.sqrt(2.), # green
... [1.,1.,1.]/np.sqrt(3.)]).T), # blue
... 'hexagonal' : np.linalg.inv(np.array([[0.,0.,1.], # direction of red
... [1.,0.,0.], # green
... [np.sqrt(3.),1.,0.]/np.sqrt(4.)]).T), # blue
... 'tetragonal' : np.linalg.inv(np.array([[0.,0.,1.], # direction of red
... [1.,0.,0.], # green
... [1.,1.,0.]/np.sqrt(2.)]).T), # blue
... 'orthorhombic': np.linalg.inv(np.array([[0.,0.,1.], # direction of red
... [1.,0.,0.], # green
... [0.,1.,0.]]).T), # blue
... }
"""
vector_ = np.array(vector)
if vector_.shape[-1] != 3:
raise ValueError('Input is not a field of three-dimensional vectors.')
if self.family == 'cubic':
basis = {'improper':np.array([ [-1. , 0. , 1. ],
[ np.sqrt(2.) , -np.sqrt(2.) , 0. ],
[ 0. , np.sqrt(3.) , 0. ] ]),
'proper':np.array([ [ 0. , -1. , 1. ],
[-np.sqrt(2.) , np.sqrt(2.) , 0. ],
[ np.sqrt(3.) , 0. , 0. ] ]),
}
elif self.family == 'hexagonal':
basis = {'improper':np.array([ [ 0. , 0. , 1. ],
[ 1. , -np.sqrt(3.) , 0. ],
[ 0. , 2. , 0. ] ]),
'proper':np.array([ [ 0. , 0. , 1. ],
[-1. , np.sqrt(3.) , 0. ],
[ np.sqrt(3.) , -1. , 0. ] ]),
}
elif self.family == 'tetragonal':
basis = {'improper':np.array([ [ 0. , 0. , 1. ],
[ 1. , -1. , 0. ],
[ 0. , np.sqrt(2.) , 0. ] ]),
'proper':np.array([ [ 0. , 0. , 1. ],
[-1. , 1. , 0. ],
[ np.sqrt(2.) , 0. , 0. ] ]),
}
elif self.family == 'orthorhombic':
basis = {'improper':np.array([ [ 0., 0., 1.],
[ 1., 0., 0.],
[ 0., 1., 0.] ]),
'proper':np.array([ [ 0., 0., 1.],
[-1., 0., 0.],
[ 0., 1., 0.] ]),
}
else: # direct exit for unspecified symmetry
return np.zeros_like(vector_)
if proper:
components_proper = np.around(np.einsum('...ji,...i',
np.broadcast_to(basis['proper'], vector_.shape+(3,)),
vector_), 12)
components_improper = np.around(np.einsum('...ji,...i',
np.broadcast_to(basis['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(basis['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
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 is None or other.family is None:
raise ValueError('Missing crystal symmetry')
if self.family != other.family:
raise NotImplementedError('Disorientation between different crystal families not supported yet.')
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, J. Appl. Cryst. 34:280-288, 2001
"Orientation distribution analysis in deformed grains"
https://doi.org/10.1107/S0021889801003077
"""
if self.family is None:
raise ValueError('Missing crystal symmetry')
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.
"""
if self.family is None:
raise ValueError('Missing crystal symmetry')
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,))
)
@classmethod
def Bravais_to_Miller(cls,uvtw=None,hkil=None):
"""
Transform 4 MillerBravais indices to 3 Miller indices of crystal direction [uvw] or plane normal (hkl).
Parameters
----------
uvtw|hkil : numpy.ndarray of shape (...,4)
MillerBravais indices of crystallographic direction [uvtw] or plane normal (hkil).
Returns
-------
uvw|hkl : numpy.ndarray of shape (...,3)
Miller indices of [uvw] direction or (hkl) plane normal.
"""
if (uvtw is not None) ^ (hkil is None):
raise KeyError('Specify either "uvtw" or "hkil"')
axis,basis = (np.array(uvtw),np.array([[1,0,-1,0],
[0,1,-1,0],
[0,0, 0,1]])) \
if hkil is None else \
(np.array(hkil),np.array([[1,0,0,0],
[0,1,0,0],
[0,0,0,1]]))
return np.einsum('il,...l->...i',basis,axis)
@classmethod
def Miller_to_Bravais(cls,uvw=None,hkl=None):
"""
Transform 3 Miller indices to 4 MillerBravais indices of crystal direction [uvtw] or plane normal (hkil).
Parameters
----------
uvw|hkl : numpy.ndarray of shape (...,3)
Miller indices of crystallographic direction [uvw] or plane normal (hkl).
Returns
-------
uvtw|hkil : numpy.ndarray of shape (...,4)
MillerBravais indices of [uvtw] direction or (hkil) plane normal.
"""
if (uvw is not None) ^ (hkl is None):
raise KeyError('Specify either "uvw" or "hkl"')
axis,basis = (np.array(uvw),np.array([[ 2,-1, 0],
[-1, 2, 0],
[-1,-1, 0],
[ 0, 0, 3]])/3) \
if hkl is None else \
(np.array(hkl),np.array([[ 1, 0, 0],
[ 0, 1, 0],
[-1,-1, 0],
[ 0, 0, 1]]))
return np.einsum('il,...l->...i',basis,axis)
def to_lattice(self,direction=None,plane=None):
"""
Calculate lattice vector corresponding to crystal frame direction or plane normal.
Parameters
----------
direction|normal : numpy.ndarray of shape (...,3)
Vector along direction or plane normal.
Returns
-------
Miller : numpy.ndarray of shape (...,3)
lattice vector of direction or plane.
Use util.scale_to_coprime to convert to (integer) Miller indices.
"""
if (direction is not None) ^ (plane is None):
raise KeyError('Specify either "direction" or "plane"')
axis,basis = (np.array(direction),self.basis_reciprocal.T) \
if plane is None else \
(np.array(plane),self.basis_real.T)
return np.einsum('il,...l->...i',basis,axis)
def to_frame(self,uvw=None,hkl=None,with_symmetry=False):
"""
Calculate crystal 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)
Crystal frame vector (or vectors if with_symmetry) along [uvw] direction or (hkl) plane normal.
"""
if (uvw is not None) ^ (hkl is None):
raise KeyError('Specify either "uvw" or "hkl"')
axis,basis = (np.array(uvw),self.basis_real) \
if hkl is None else \
(np.array(hkl),self.basis_reciprocal)
return (self.symmetry_operations.broadcast_to(self.symmetry_operations.shape+axis.shape[:-1],mode='right')
@ np.broadcast_to(np.einsum('il,...l->...i',basis,axis),self.symmetry_operations.shape+axis.shape)
if with_symmetry else
np.einsum('il,...l->...i',basis,axis))
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,with_symmetry=with_symmetry)
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 : str
Type of kinematics, i.e. 'slip' or 'twin'.
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 slip systems 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'],with_symmetry=False)
p = self.to_frame(hkl=self.kinematics[mode]['plane'] ,with_symmetry=False)
P = np.einsum('...i,...j->...ij',d/np.linalg.norm(d,axis=-1,keepdims=True),
p/np.linalg.norm(p,axis=-1,keepdims=True))
return ~self.broadcast_to( self.shape+P.shape[:-2],mode='right') \
@ np.broadcast_to(P,self.shape+P.shape)