import copy from typing import Optional, Union, TypeVar import numpy as np from ._typehints import FloatSequence, IntSequence, CrystalFamily, BravaisLattice from . import Rotation from . import Crystal from . import util from . import tensor MyType = TypeVar('MyType', bound='Orientation') 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),family='tetragonal').reduced """ @util.extend_docstring(adopted_parameters=Crystal.__init__) def __init__(self, rotation: Union[FloatSequence, Rotation] = np.array([1.,0.,0.,0.]), *, family: Optional[CrystalFamily] = None, lattice: Optional[BravaisLattice] = None, a: Optional[float] = None, b: Optional[float] = None, c: Optional[float] = None, alpha: Optional[float] = None, beta: Optional[float] = None, gamma: Optional[float] = None, degrees: bool = False): """ New orientation. Parameters ---------- rotation : list, numpy.ndarray, or 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) -> str: """ Return repr(self). Give short, human-readable summary. """ return util.srepr([Crystal.__repr__(self), Rotation.__repr__(self)]) def __copy__(self: MyType, rotation: Union[None, FloatSequence, Rotation] = None) -> MyType: """ Return deepcopy(self). Create deep copy. """ dup = copy.deepcopy(self) if rotation is not None: dup.quaternion = Rotation(rotation).quaternion return dup copy = __copy__ def __eq__(self, other: object) -> bool: """ Return self==other. Test equality of other. Parameters ---------- other : Orientation Orientation to check for equality. """ if not isinstance(other, Orientation): return NotImplemented 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: object) -> bool: """ Return self!=other. Test inequality of other. Parameters ---------- other : Orientation Orientation to check for equality. """ return np.logical_not(self==other) if isinstance(other, Orientation) else NotImplemented def isclose(self: MyType, other: MyType, rtol: float = 1e-5, atol: float = 1e-8, equal_nan: bool = True) -> bool: """ 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 of bool, shape (self.shape) 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: MyType, other: MyType, rtol: float = 1e-5, atol: float = 1e-8, equal_nan: bool = True) -> bool: """ 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 bool(np.all(self.isclose(other,rtol,atol,equal_nan))) def __mul__(self: MyType, other: Union[Rotation, 'Orientation']) -> MyType: """ Return self*other. Compose with other. Parameters ---------- other : Rotation or Orientation, shape (self.shape) Object for composition. Returns ------- composition : Orientation Compound rotation self*other, i.e. first other then self rotation. """ if not isinstance(other, (Orientation,Rotation)): raise TypeError('use "O@b", i.e. matmul, to apply Orientation "O" to object "b"') return self.copy(Rotation(self.quaternion)*Rotation(other.quaternion)) @classmethod @util.extend_docstring(Rotation.from_random, adopted_parameters=Crystal.__init__) @util.pass_on('rotation', Rotation.from_random, wrapped=__init__) def from_random(cls, **kwargs) -> 'Orientation': return cls(**kwargs) @classmethod @util.extend_docstring(Rotation.from_quaternion, adopted_parameters=Crystal.__init__) @util.pass_on('rotation', Rotation.from_quaternion, wrapped=__init__) def from_quaternion(cls, **kwargs) -> 'Orientation': return cls(**kwargs) @classmethod @util.extend_docstring(Rotation.from_Euler_angles, adopted_parameters=Crystal.__init__) @util.pass_on('rotation', Rotation.from_Euler_angles, wrapped=__init__) def from_Euler_angles(cls, **kwargs) -> 'Orientation': return cls(**kwargs) @classmethod @util.extend_docstring(Rotation.from_axis_angle, adopted_parameters=Crystal.__init__) @util.pass_on('rotation', Rotation.from_axis_angle, wrapped=__init__) def from_axis_angle(cls, **kwargs) -> 'Orientation': return cls(**kwargs) @classmethod @util.extend_docstring(Rotation.from_basis, adopted_parameters=Crystal.__init__) @util.pass_on('rotation', Rotation.from_basis, wrapped=__init__) def from_basis(cls, **kwargs) -> 'Orientation': return cls(**kwargs) @classmethod @util.extend_docstring(Rotation.from_matrix, adopted_parameters=Crystal.__init__) @util.pass_on('rotation', Rotation.from_matrix, wrapped=__init__) def from_matrix(cls, **kwargs) -> 'Orientation': return cls(**kwargs) @classmethod @util.extend_docstring(Rotation.from_Rodrigues_vector, adopted_parameters=Crystal.__init__) @util.pass_on('rotation', Rotation.from_Rodrigues_vector, wrapped=__init__) def from_Rodrigues_vector(cls, **kwargs) -> 'Orientation': return cls(**kwargs) @classmethod @util.extend_docstring(Rotation.from_homochoric, adopted_parameters=Crystal.__init__) @util.pass_on('rotation', Rotation.from_homochoric, wrapped=__init__) def from_homochoric(cls, **kwargs) -> 'Orientation': return cls(**kwargs) @classmethod @util.extend_docstring(Rotation.from_cubochoric, adopted_parameters=Crystal.__init__) @util.pass_on('rotation', Rotation.from_cubochoric, wrapped=__init__) def from_cubochoric(cls, **kwargs) -> 'Orientation': return cls(**kwargs) @classmethod @util.extend_docstring(Rotation.from_spherical_component, adopted_parameters=Crystal.__init__) @util.pass_on('rotation', Rotation.from_spherical_component, wrapped=__init__) def from_spherical_component(cls, **kwargs) -> 'Orientation': return cls(**kwargs) @classmethod @util.extend_docstring(Rotation.from_fiber_component, adopted_parameters=Crystal.__init__) @util.pass_on('rotation', Rotation.from_fiber_component, wrapped=__init__) def from_fiber_component(cls, **kwargs) -> 'Orientation': return cls(**kwargs) @classmethod @util.extend_docstring(adopted_parameters=Crystal.__init__) def from_directions(cls, uvw: FloatSequence, hkl: FloatSequence, **kwargs) -> 'Orientation': """ Initialize orientation object from two crystallographic directions. Parameters ---------- uvw : numpy.ndarray, shape (...,3) Lattice direction aligned with lab frame x-direction. hkl : numpy.ndarray, shape (...,3) Lattice plane normal aligned with lab frame z-direction. Returns ------- new : damask.Orientation """ 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.from_matrix(tensor.transpose(om/np.linalg.norm(om,axis=-1,keepdims=True)))) @property def equivalent(self: MyType) -> MyType: """ 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(o*Rotation(self.quaternion).broadcast_to(o.shape,mode='left')) @property def reduced(self: MyType) -> MyType: """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) -> Union[np.bool_, np.ndarray]: """ Check whether orientation falls into fundamental zone of own symmetry. Returns ------- in : numpy.ndarray of bool, shape (self.shape) 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) if 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) if 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) if self.family == 'orthorhombic': return (np.prod(1. >= rho_abs,axis=-1)).astype(bool) if self.family == 'monoclinic': return np.logical_or( 1. >= rho_abs[...,1], np.isnan(rho_abs[...,1])) if self.family == 'triclinic': return np.ones(rho_abs.shape[:-1]).astype(bool) raise TypeError(f'unknown symmetry "{self.family}"') @property def in_disorientation_FZ(self) -> np.ndarray: """ Check whether orientation falls into fundamental zone of disorientations. Returns ------- in : numpy.ndarray of bool, shape (self.shape) 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 """ def larger_or_equal(v,c): return ((np.isclose(c[0],v[...,0]) | (v[...,0] > c[0])) & (np.isclose(c[1],v[...,1]) | (v[...,1] > c[1])) & (np.isclose(c[2],v[...,2]) | (v[...,2] > c[2]))).astype(bool) rho = self.as_Rodrigues_vector(compact=True) return larger_or_equal(rho, [rho[...,1], rho[...,2],0] if self.family == 'cubic' else [rho[...,1]*np.sqrt(3),0, 0] if self.family == 'hexagonal' else [rho[...,1], 0, 0] if self.family == 'tetragonal' else [0, 0, 0] if self.family == 'orthorhombic' else [-np.inf, 0, 0] if self.family == 'monoclinic' else [-np.inf, -np.inf, -np.inf]) & self.in_FZ def disorientation(self, other: 'Orientation', return_operators: bool = False) -> object: """ Calculate disorientation between self 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 of int, shape (...,2), conditional Index of symmetrically equivalent orientation that rotated vector to the SST. Notes ----- Requires same crystal family for both orientations. Examples -------- Disorientation between two specific orientations of hexagonal symmetry: >>> import damask >>> a = damask.Orientation.from_Euler_angles(phi=[123,32,21],degrees=True,family='hexagonal') >>> b = damask.Orientation.from_Euler_angles(phi=[104,11,87],degrees=True,family='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) Plot a sample from the Mackenzie distribution. >>> import matplotlib.pyplot as plt >>> import damask >>> N = 10000 >>> a = damask.Orientation.from_random(shape=N,family='cubic') >>> b = damask.Orientation.from_random(shape=N,family='cubic') >>> n,omega = a.disorientation(b).as_axis_angle(degrees=True,pair=True) >>> plt.hist(omega,25) >>> plt.show() """ # For extension to cases with differing symmetry see # https://doi.org/10.1107/S0021889808016373 and https://doi.org/10.1107/S0108767391006864 if self.family != other.family: raise NotImplementedError('disorientation between different crystal families') blend = util.shapeblender( self.shape,other.shape) s_m = util.shapeshifter( self.shape,blend,mode='right') s_o = util.shapeshifter(other.shape,blend,mode='left') s = self.broadcast_to(s_m).equivalent o = other.broadcast_to(s_o).equivalent r_ = s[:,np.newaxis,...].misorientation(o[np.newaxis,:,...]) # type: ignore[index] _r = ~r_ forward = r_.in_disorientation_FZ reverse = _r.in_disorientation_FZ ok = forward | reverse ok &= (np.cumsum(ok.reshape((-1,)+blend),axis=0) == 1).reshape(ok.shape) r = np.where(np.any((ok&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: Optional[FloatSequence] = None, return_cloud: bool = False): """ Return orientation average over last dimension. Parameters ---------- weights : numpy.ndarray, shape (self.shape), optional Relative weights of orientations. Defaults to equal weights. return_cloud : bool, optional Return the specific (symmetrically equivalent) orientations that were averaged. Defaults to False. Returns ------- average : Orientation Weighted average of original Orientation field. cloud : Orientations, conditional Symmetrically equivalent version of each orientation that were actually 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,)) # type: ignore .broadcast_to(eq.shape)).as_axis_angle()[...,3] # type: ignore 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(r).average(weights)),self.copy(Rotation(r))) if return_cloud else self.copy(Rotation(r).average(weights)) ) def to_SST(self, vector: FloatSequence, proper: bool = False, return_operators: bool = False) -> np.ndarray: """ Rotate lab frame vector to ensure it falls into (improper or proper) standard stereographic triangle of crystal symmetry. Parameters ---------- vector : numpy.ndarray, 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 vector array of shape (2,4) 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, shape (...,3) Rotated vector falling into SST. operator : numpy.ndarray of int, shape (...), conditional Index of symmetrically equivalent orientation that rotated vector to SST. """ vector_ = np.array(vector,float) if vector_.shape[-1] != 3: raise ValueError('input is not a field of three-dimensional vectors') blend = util.shapeblender( self.shape,vector_.shape[:-1]) eq = self.broadcast_to(util.shapeshifter( self.shape,blend,mode='right')).equivalent poles = np.atleast_2d(eq @ np.broadcast_to(vector_,(1,)+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+(3,)), (np.vstack(loc[:1]).T)[sort].reshape(blend)) if return_operators else poles[ok][sort].reshape(blend+(3,)) ) def in_SST(self, vector: FloatSequence, proper: bool = False) -> Union[np.bool_, np.ndarray]: """ Check whether given crystal frame vector falls into standard stereographic triangle of own symmetry. Parameters ---------- vector : numpy.ndarray, 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. """ vector_ = np.array(vector,float) if 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: FloatSequence, in_SST: bool = True, proper: bool = False) -> np.ndarray: """ Map lab frame vector to RGB color within standard stereographic triangle of own symmetry. Parameters ---------- vector : numpy.ndarray, shape (...,3) Lab frame vector to colorize. Shape of vector blends with shape of own rotation array. For example, a rotation array of shape (3,2) and a vector array of shape (2,4) 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, shape (...,3) RGB array of IPF colors. Examples -------- Inverse pole figure color of the e_3 lab 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.]) Sample standard triangle for hexagonal symmetry: >>> import damask >>> from matplotlib import pyplot as plt >>> lab = [0,0,1] >>> o = damask.Orientation.from_random(shape=500000,family='hexagonal') >>> coord = damask.util.project_equal_area(o.to_SST(lab)) >>> color = o.IPF_color(lab) >>> plt.scatter(coord[:,0],coord[:,1],color=color,s=.06) >>> plt.axis('scaled') >>> plt.show() """ 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))**(1./3.) # 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 #################################################################################################### # functions that require lattice, not just family def to_pole(self, *, uvw: Optional[FloatSequence] = None, hkl: Optional[FloatSequence] = None, with_symmetry: bool = False, normalize: bool = True) -> np.ndarray: """ Calculate lab frame vector along lattice direction [uvw] or plane normal (hkl). Parameters ---------- uvw|hkl : numpy.ndarray, 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 vector array of shape (2,4) result in (3,2,4) outputs. with_symmetry : bool, optional Calculate all N symmetrically equivalent vectors. Defaults to False. normalize : bool, optional Normalize output vector. Defaults to True. Returns ------- vector : numpy.ndarray, 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) s_v = v.shape[:-1] blend = util.shapeblender(self.shape,s_v) if normalize: v /= np.linalg.norm(v,axis=-1,keepdims=len(s_v)>0) if with_symmetry: sym_ops = self.symmetry_operations s_v += sym_ops.shape blend += sym_ops.shape v = sym_ops.broadcast_to(s_v) @ v[...,np.newaxis,:] return ~(self.broadcast_to(blend)) @ np.broadcast_to(v,blend+(3,)) def Schmid(self, *, N_slip: Optional[IntSequence] = None, N_twin: Optional[IntSequence] = None) -> np.ndarray: u""" Calculate Schmid matrix P = d ⨂ n in the lab frame for selected deformation systems. Parameters ---------- N_slip|N_twin : '*' or sequence of int Number of deformation systems per family of the deformation system. Use '*' to select all. Returns ------- P : numpy.ndarray, 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 numpy as np >>> import damask >>> 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=[12])[0] 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']] if not active: raise ValueError('Schmid matrix not defined') 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: MyType, model: str) -> MyType: """ All orientations related to self by given relationship model. Parameters ---------- model : str Orientation relationship model selected from self.orientation_relationships. Returns ------- rel : Orientation, shape (:,self.shape) Orientations related to self according to the selected model for the orientation relationship. Examples -------- Face-centered cubic orientations following from a body-centered cubic crystal in "Cube" orientation according to the Bain orientation relationship (cI -> cF). >>> import numpy as np >>> import damask >>> np.set_printoptions(3,suppress=True,floatmode='fixed') >>> damask.Orientation(lattice='cI').related('Bain') Crystal family: cubic Bravais lattice: cF a=1 m, b=1 m, c=1 m α=90°, β=90°, γ=90° Quaternions of shape (3,) [[0.924 0.383 0.000 0.000] [0.924 0.000 0.383 0.000] [0.924 0.000 0.000 0.383]] """ 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, )