import numpy as np from . import util from . import LatticeFamily from . import Rotation 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', } class Crystal(LatticeFamily): """Lattice.""" def __init__(self,*, family = None, lattice = None, a = None,b = None,c = None, alpha = None,beta = None,gamma = None, degrees = False): """ Lattice. Parameters ---------- lattice : {'aP', 'mP', 'mS', 'oP', 'oS', 'oI', 'oF', 'tP', 'tI', 'hP', 'cP', 'cI', 'cF'}. 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. """ super().__init__(family = lattice_symmetries[lattice] if family is None else family) self.lattice = lattice if self.lattice is not None: self.a = 1 if a is None else a self.b = b self.c = c 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 = np.radians(alpha) if degrees and alpha is not None else alpha self.beta = np.radians(beta) if degrees and beta is not None else beta self.gamma = np.radians(gamma) if degrees and gamma is not None else gamma if self.alpha is None and 'alpha' in self.immutable: self.alpha = self.immutable['alpha'] if self.beta is None and 'beta' in self.immutable: self.beta = self.immutable['beta'] if self.gamma is None and 'gamma' in self.immutable: self.gamma = self.immutable['gamma'] 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') else: self.a = self.b = self.c = None self.alpha = self.beta = self.gamma = None def __eq__(self,other): """ Equal to other. Parameters ---------- other : Lattice Lattice to check for equality. """ return self.lattice == other.lattice and self.parameters == other.parameters @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 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): """ Return orthogonal real space crystal basis. References ---------- C.T. Young and J.L. Lytton, Journal of Applied Physics 43:1408–1417, 1972 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): """Return reciprocal (dual) crystal basis.""" return np.linalg.inv(self.basis_real.T) def to_lattice(self,*,direction=None,plane=None): """ Calculate lattice vector corresponding to crystal frame direction or plane normal. Parameters ---------- direction|plane : 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',basis,axis) def to_frame(self,*,uvw=None,hkl=None): """ 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. 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 np.einsum('il,...l',basis,axis) def kinematics(self,mode): master = self._kinematics[self.lattice][mode] if self.lattice == 'hP': return {'direction':util.Bravais_to_Miller(uvtw=master[:,0:4]), 'plane': util.Bravais_to_Miller(hkil=master[:,4:8])} else: return {'direction':master[:,0:3], 'plane': master[:,3:6]} def relation_operations(self,model): """ Crystallographic orientation relationships for phase transformations. Parameters ---------- model : str Name of orientation relationship. Returns ------- operations : (string, damask.Rotation) 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 """ if model not in self.orientation_relationships: raise KeyError(f'unknown orientation relationship "{model}"') r = self.orientation_relationships[model] 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 util.Bravais_to_Miller(uvtw=m[...,0,0:4]) p_[...,1,:] = m[...,1,:] if m.shape[-1] == 3 else util.Bravais_to_Miller(hkil=m[...,1,0:4]) _p[...,0,:] = o[...,0,:] if o.shape[-1] == 3 else util.Bravais_to_Miller(uvtw=o[...,0,0:4]) _p[...,1,:] = o[...,1,:] if o.shape[-1] == 3 else util.Bravais_to_Miller(hkil=o[...,1,0:4]) return (ol,Rotation.from_parallel(p_,_p)) @property def orientation_relationships(self): return {k:v for k,v in self._orientation_relationships.items() if self.lattice in v} _kinematics = { 'cF': { 'slip' : np.array([ [+0,+1,-1, +1,+1,+1], [-1,+0,+1, +1,+1,+1], [+1,-1,+0, +1,+1,+1], [+0,-1,-1, -1,-1,+1], [+1,+0,+1, -1,-1,+1], [-1,+1,+0, -1,-1,+1], [+0,-1,+1, +1,-1,-1], [-1,+0,-1, +1,-1,-1], [+1,+1,+0, +1,-1,-1], [+0,+1,+1, -1,+1,-1], [+1,+0,-1, -1,+1,-1], [-1,-1,+0, -1,+1,-1], [+1,+1,+0, +1,-1,+0], [+1,-1,+0, +1,+1,+0], [+1,+0,+1, +1,+0,-1], [+1,+0,-1, +1,+0,+1], [+0,+1,+1, +0,+1,-1], [+0,+1,-1, +0,+1,+1], ],'d'), 'twin' : np.array([ [-2, 1, 1, 1, 1, 1], [ 1,-2, 1, 1, 1, 1], [ 1, 1,-2, 1, 1, 1], [ 2,-1, 1, -1,-1, 1], [-1, 2, 1, -1,-1, 1], [-1,-1,-2, -1,-1, 1], [-2,-1,-1, 1,-1,-1], [ 1, 2,-1, 1,-1,-1], [ 1,-1, 2, 1,-1,-1], [ 2, 1,-1, -1, 1,-1], [-1,-2,-1, -1, 1,-1], [-1, 1, 2, -1, 1,-1], ],dtype=float), }, 'cI': { 'slip' : np.array([ [+1,-1,+1, +0,+1,+1], [-1,-1,+1, +0,+1,+1], [+1,+1,+1, +0,-1,+1], [-1,+1,+1, +0,-1,+1], [-1,+1,+1, +1,+0,+1], [-1,-1,+1, +1,+0,+1], [+1,+1,+1, -1,+0,+1], [+1,-1,+1, -1,+0,+1], [-1,+1,+1, +1,+1,+0], [-1,+1,-1, +1,+1,+0], [+1,+1,+1, -1,+1,+0], [+1,+1,-1, -1,+1,+0], [-1,+1,+1, +2,+1,+1], [+1,+1,+1, -2,+1,+1], [+1,+1,-1, +2,-1,+1], [+1,-1,+1, +2,+1,-1], [+1,-1,+1, +1,+2,+1], [+1,+1,-1, -1,+2,+1], [+1,+1,+1, +1,-2,+1], [-1,+1,+1, +1,+2,-1], [+1,+1,-1, +1,+1,+2], [+1,-1,+1, -1,+1,+2], [-1,+1,+1, +1,-1,+2], [+1,+1,+1, +1,+1,-2], [+1,+1,-1, +1,+2,+3], [+1,-1,+1, -1,+2,+3], [-1,+1,+1, +1,-2,+3], [+1,+1,+1, +1,+2,-3], [+1,-1,+1, +1,+3,+2], [+1,+1,-1, -1,+3,+2], [+1,+1,+1, +1,-3,+2], [-1,+1,+1, +1,+3,-2], [+1,+1,-1, +2,+1,+3], [+1,-1,+1, -2,+1,+3], [-1,+1,+1, +2,-1,+3], [+1,+1,+1, +2,+1,-3], [+1,-1,+1, +2,+3,+1], [+1,+1,-1, -2,+3,+1], [+1,+1,+1, +2,-3,+1], [-1,+1,+1, +2,+3,-1], [-1,+1,+1, +3,+1,+2], [+1,+1,+1, -3,+1,+2], [+1,+1,-1, +3,-1,+2], [+1,-1,+1, +3,+1,-2], [-1,+1,+1, +3,+2,+1], [+1,+1,+1, -3,+2,+1], [+1,+1,-1, +3,-2,+1], [+1,-1,+1, +3,+2,-1], ],'d'), 'twin' : np.array([ [-1, 1, 1, 2, 1, 1], [ 1, 1, 1, -2, 1, 1], [ 1, 1,-1, 2,-1, 1], [ 1,-1, 1, 2, 1,-1], [ 1,-1, 1, 1, 2, 1], [ 1, 1,-1, -1, 2, 1], [ 1, 1, 1, 1,-2, 1], [-1, 1, 1, 1, 2,-1], [ 1, 1,-1, 1, 1, 2], [ 1,-1, 1, -1, 1, 2], [-1, 1, 1, 1,-1, 2], [ 1, 1, 1, 1, 1,-2], ],dtype=float), }, 'hP': { 'slip' : np.array([ [+2,-1,-1,+0, +0,+0,+0,+1], [-1,+2,-1,+0, +0,+0,+0,+1], [-1,-1,+2,+0, +0,+0,+0,+1], [+2,-1,-1,+0, +0,+1,-1,+0], [-1,+2,-1,+0, -1,+0,+1,+0], [-1,-1,+2,+0, +1,-1,+0,+0], [-1,+1,+0,+0, +1,+1,-2,+0], [+0,-1,+1,+0, -2,+1,+1,+0], [+1,+0,-1,+0, +1,-2,+1,+0], [-1,+2,-1,+0, +1,+0,-1,+1], [-2,+1,+1,+0, +0,+1,-1,+1], [-1,-1,+2,+0, -1,+1,+0,+1], [+1,-2,+1,+0, -1,+0,+1,+1], [+2,-1,-1,+0, +0,-1,+1,+1], [+1,+1,-2,+0, +1,-1,+0,+1], [-2,+1,+1,+3, +1,+0,-1,+1], [-1,-1,+2,+3, +1,+0,-1,+1], [-1,-1,+2,+3, +0,+1,-1,+1], [+1,-2,+1,+3, +0,+1,-1,+1], [+1,-2,+1,+3, -1,+1,+0,+1], [+2,-1,-1,+3, -1,+1,+0,+1], [+2,-1,-1,+3, -1,+0,+1,+1], [+1,+1,-2,+3, -1,+0,+1,+1], [+1,+1,-2,+3, +0,-1,+1,+1], [-1,+2,-1,+3, +0,-1,+1,+1], [-1,+2,-1,+3, +1,-1,+0,+1], [-2,+1,+1,+3, +1,-1,+0,+1], [-1,-1,+2,+3, +1,+1,-2,+2], [+1,-2,+1,+3, -1,+2,-1,+2], [+2,-1,-1,+3, -2,+1,+1,+2], [+1,+1,-2,+3, -1,-1,+2,+2], [-1,+2,-1,+3, +1,-2,+1,+2], [-2,+1,+1,+3, +2,-1,-1,+2], ],'d'), 'twin' : np.array([ [-1, 0, 1, 1, 1, 0,-1, 2], # shear = (3-(c/a)^2)/(sqrt(3) c/a) <-10.1>{10.2} [ 0,-1, 1, 1, 0, 1,-1, 2], [ 1,-1, 0, 1, -1, 1, 0, 2], [ 1, 0,-1, 1, -1, 0, 1, 2], [ 0, 1,-1, 1, 0,-1, 1, 2], [-1, 1, 0, 1, 1,-1, 0, 2], [-1,-1, 2, 6, 1, 1,-2, 1], # shear = 1/(c/a) <11.6>{-1-1.1} [ 1,-2, 1, 6, -1, 2,-1, 1], [ 2,-1,-1, 6, -2, 1, 1, 1], [ 1, 1,-2, 6, -1,-1, 2, 1], [-1, 2,-1, 6, 1,-2, 1, 1], [-2, 1, 1, 6, 2,-1,-1, 1], [ 1, 0,-1,-2, 1, 0,-1, 1], # shear = (4(c/a)^2-9)/(4 sqrt(3) c/a) <10.-2>{10.1} [ 0, 1,-1,-2, 0, 1,-1, 1], [-1, 1, 0,-2, -1, 1, 0, 1], [-1, 0, 1,-2, -1, 0, 1, 1], [ 0,-1, 1,-2, 0,-1, 1, 1], [ 1,-1, 0,-2, 1,-1, 0, 1], [ 1, 1,-2,-3, 1, 1,-2, 2], # shear = 2((c/a)^2-2)/(3 c/a) <11.-3>{11.2} [-1, 2,-1,-3, -1, 2,-1, 2], [-2, 1, 1,-3, -2, 1, 1, 2], [-1,-1, 2,-3, -1,-1, 2, 2], [ 1,-2, 1,-3, 1,-2, 1, 2], [ 2,-1,-1,-3, 2,-1,-1, 2], ],dtype=float), }, } _orientation_relationships = { 'KS': { 'cF' : np.array([ [[-1, 0, 1],[ 1, 1, 1]], [[-1, 0, 1],[ 1, 1, 1]], [[ 0, 1,-1],[ 1, 1, 1]], [[ 0, 1,-1],[ 1, 1, 1]], [[ 1,-1, 0],[ 1, 1, 1]], [[ 1,-1, 0],[ 1, 1, 1]], [[ 1, 0,-1],[ 1,-1, 1]], [[ 1, 0,-1],[ 1,-1, 1]], [[-1,-1, 0],[ 1,-1, 1]], [[-1,-1, 0],[ 1,-1, 1]], [[ 0, 1, 1],[ 1,-1, 1]], [[ 0, 1, 1],[ 1,-1, 1]], [[ 0,-1, 1],[-1, 1, 1]], [[ 0,-1, 1],[-1, 1, 1]], [[-1, 0,-1],[-1, 1, 1]], [[-1, 0,-1],[-1, 1, 1]], [[ 1, 1, 0],[-1, 1, 1]], [[ 1, 1, 0],[-1, 1, 1]], [[-1, 1, 0],[ 1, 1,-1]], [[-1, 1, 0],[ 1, 1,-1]], [[ 0,-1,-1],[ 1, 1,-1]], [[ 0,-1,-1],[ 1, 1,-1]], [[ 1, 0, 1],[ 1, 1,-1]], [[ 1, 0, 1],[ 1, 1,-1]], ],dtype=float), 'cI' : np.array([ [[-1,-1, 1],[ 0, 1, 1]], [[-1, 1,-1],[ 0, 1, 1]], [[-1,-1, 1],[ 0, 1, 1]], [[-1, 1,-1],[ 0, 1, 1]], [[-1,-1, 1],[ 0, 1, 1]], [[-1, 1,-1],[ 0, 1, 1]], [[-1,-1, 1],[ 0, 1, 1]], [[-1, 1,-1],[ 0, 1, 1]], [[-1,-1, 1],[ 0, 1, 1]], [[-1, 1,-1],[ 0, 1, 1]], [[-1,-1, 1],[ 0, 1, 1]], [[-1, 1,-1],[ 0, 1, 1]], [[-1,-1, 1],[ 0, 1, 1]], [[-1, 1,-1],[ 0, 1, 1]], [[-1,-1, 1],[ 0, 1, 1]], [[-1, 1,-1],[ 0, 1, 1]], [[-1,-1, 1],[ 0, 1, 1]], [[-1, 1,-1],[ 0, 1, 1]], [[-1,-1, 1],[ 0, 1, 1]], [[-1, 1,-1],[ 0, 1, 1]], [[-1,-1, 1],[ 0, 1, 1]], [[-1, 1,-1],[ 0, 1, 1]], [[-1,-1, 1],[ 0, 1, 1]], [[-1, 1,-1],[ 0, 1, 1]], ],dtype=float), }, 'GT': { 'cF' : np.array([ [[ -5,-12, 17],[ 1, 1, 1]], [[ 17, -5,-12],[ 1, 1, 1]], [[-12, 17, -5],[ 1, 1, 1]], [[ 5, 12, 17],[ -1, -1, 1]], [[-17, 5,-12],[ -1, -1, 1]], [[ 12,-17, -5],[ -1, -1, 1]], [[ -5, 12,-17],[ -1, 1, 1]], [[ 17, 5, 12],[ -1, 1, 1]], [[-12,-17, 5],[ -1, 1, 1]], [[ 5,-12,-17],[ 1, -1, 1]], [[-17, -5, 12],[ 1, -1, 1]], [[ 12, 17, 5],[ 1, -1, 1]], [[ -5, 17,-12],[ 1, 1, 1]], [[-12, -5, 17],[ 1, 1, 1]], [[ 17,-12, -5],[ 1, 1, 1]], [[ 5,-17,-12],[ -1, -1, 1]], [[ 12, 5, 17],[ -1, -1, 1]], [[-17, 12, -5],[ -1, -1, 1]], [[ -5,-17, 12],[ -1, 1, 1]], [[-12, 5,-17],[ -1, 1, 1]], [[ 17, 12, 5],[ -1, 1, 1]], [[ 5, 17, 12],[ 1, -1, 1]], [[ 12, -5,-17],[ 1, -1, 1]], [[-17,-12, 5],[ 1, -1, 1]], ],dtype=float), 'cI' : np.array([ [[-17, -7, 17],[ 1, 0, 1]], [[ 17,-17, -7],[ 1, 1, 0]], [[ -7, 17,-17],[ 0, 1, 1]], [[ 17, 7, 17],[ -1, 0, 1]], [[-17, 17, -7],[ -1, -1, 0]], [[ 7,-17,-17],[ 0, -1, 1]], [[-17, 7,-17],[ -1, 0, 1]], [[ 17, 17, 7],[ -1, 1, 0]], [[ -7,-17, 17],[ 0, 1, 1]], [[ 17, -7,-17],[ 1, 0, 1]], [[-17,-17, 7],[ 1, -1, 0]], [[ 7, 17, 17],[ 0, -1, 1]], [[-17, 17, -7],[ 1, 1, 0]], [[ -7,-17, 17],[ 0, 1, 1]], [[ 17, -7,-17],[ 1, 0, 1]], [[ 17,-17, -7],[ -1, -1, 0]], [[ 7, 17, 17],[ 0, -1, 1]], [[-17, 7,-17],[ -1, 0, 1]], [[-17,-17, 7],[ -1, 1, 0]], [[ -7, 17,-17],[ 0, 1, 1]], [[ 17, 7, 17],[ -1, 0, 1]], [[ 17, 17, 7],[ 1, -1, 0]], [[ 7,-17,-17],[ 0, -1, 1]], [[-17, -7, 17],[ 1, 0, 1]], ],dtype=float), }, 'GT_prime': { 'cF' : np.array([ [[ 0, 1, -1],[ 7, 17, 17]], [[ -1, 0, 1],[ 17, 7, 17]], [[ 1, -1, 0],[ 17, 17, 7]], [[ 0, -1, -1],[ -7,-17, 17]], [[ 1, 0, 1],[-17, -7, 17]], [[ 1, -1, 0],[-17,-17, 7]], [[ 0, 1, -1],[ 7,-17,-17]], [[ 1, 0, 1],[ 17, -7,-17]], [[ -1, -1, 0],[ 17,-17, -7]], [[ 0, -1, -1],[ -7, 17,-17]], [[ -1, 0, 1],[-17, 7,-17]], [[ -1, -1, 0],[-17, 17, -7]], [[ 0, -1, 1],[ 7, 17, 17]], [[ 1, 0, -1],[ 17, 7, 17]], [[ -1, 1, 0],[ 17, 17, 7]], [[ 0, 1, 1],[ -7,-17, 17]], [[ -1, 0, -1],[-17, -7, 17]], [[ -1, 1, 0],[-17,-17, 7]], [[ 0, -1, 1],[ 7,-17,-17]], [[ -1, 0, -1],[ 17, -7,-17]], [[ 1, 1, 0],[ 17,-17, -7]], [[ 0, 1, 1],[ -7, 17,-17]], [[ 1, 0, -1],[-17, 7,-17]], [[ 1, 1, 0],[-17, 17, -7]], ],dtype=float), 'cI' : np.array([ [[ 1, 1, -1],[ 12, 5, 17]], [[ -1, 1, 1],[ 17, 12, 5]], [[ 1, -1, 1],[ 5, 17, 12]], [[ -1, -1, -1],[-12, -5, 17]], [[ 1, -1, 1],[-17,-12, 5]], [[ 1, -1, -1],[ -5,-17, 12]], [[ -1, 1, -1],[ 12, -5,-17]], [[ 1, 1, 1],[ 17,-12, -5]], [[ -1, -1, 1],[ 5,-17,-12]], [[ 1, -1, -1],[-12, 5,-17]], [[ -1, -1, 1],[-17, 12, -5]], [[ -1, -1, -1],[ -5, 17,-12]], [[ 1, -1, 1],[ 12, 17, 5]], [[ 1, 1, -1],[ 5, 12, 17]], [[ -1, 1, 1],[ 17, 5, 12]], [[ -1, 1, 1],[-12,-17, 5]], [[ -1, -1, -1],[ -5,-12, 17]], [[ -1, 1, -1],[-17, -5, 12]], [[ -1, -1, 1],[ 12,-17, -5]], [[ -1, 1, -1],[ 5,-12,-17]], [[ 1, 1, 1],[ 17, -5,-12]], [[ 1, 1, 1],[-12, 17, -5]], [[ 1, -1, -1],[ -5, 12,-17]], [[ 1, 1, -1],[-17, 5,-12]], ],dtype=float), }, 'NW': { 'cF' : np.array([ [[ 2, -1, -1],[ 1, 1, 1]], [[ -1, 2, -1],[ 1, 1, 1]], [[ -1, -1, 2],[ 1, 1, 1]], [[ -2, -1, -1],[ -1, 1, 1]], [[ 1, 2, -1],[ -1, 1, 1]], [[ 1, -1, 2],[ -1, 1, 1]], [[ 2, 1, -1],[ 1, -1, 1]], [[ -1, -2, -1],[ 1, -1, 1]], [[ -1, 1, 2],[ 1, -1, 1]], [[ 2, -1, 1],[ -1, -1, 1]], [[ -1, 2, 1],[ -1, -1, 1]], [[ -1, -1, -2],[ -1, -1, 1]], ],dtype=float), 'cI' : np.array([ [[ 0, -1, 1],[ 0, 1, 1]], [[ 0, -1, 1],[ 0, 1, 1]], [[ 0, -1, 1],[ 0, 1, 1]], [[ 0, -1, 1],[ 0, 1, 1]], [[ 0, -1, 1],[ 0, 1, 1]], [[ 0, -1, 1],[ 0, 1, 1]], [[ 0, -1, 1],[ 0, 1, 1]], [[ 0, -1, 1],[ 0, 1, 1]], [[ 0, -1, 1],[ 0, 1, 1]], [[ 0, -1, 1],[ 0, 1, 1]], [[ 0, -1, 1],[ 0, 1, 1]], [[ 0, -1, 1],[ 0, 1, 1]], ],dtype=float), }, 'Pitsch': { 'cF' : np.array([ [[ 1, 0, 1],[ 0, 1, 0]], [[ 1, 1, 0],[ 0, 0, 1]], [[ 0, 1, 1],[ 1, 0, 0]], [[ 0, 1, -1],[ 1, 0, 0]], [[ -1, 0, 1],[ 0, 1, 0]], [[ 1, -1, 0],[ 0, 0, 1]], [[ 1, 0, -1],[ 0, 1, 0]], [[ -1, 1, 0],[ 0, 0, 1]], [[ 0, -1, 1],[ 1, 0, 0]], [[ 0, 1, 1],[ 1, 0, 0]], [[ 1, 0, 1],[ 0, 1, 0]], [[ 1, 1, 0],[ 0, 0, 1]], ],dtype=float), 'cI' : np.array([ [[ 1, -1, 1],[ -1, 0, 1]], [[ 1, 1, -1],[ 1, -1, 0]], [[ -1, 1, 1],[ 0, 1, -1]], [[ -1, 1, -1],[ 0, -1, -1]], [[ -1, -1, 1],[ -1, 0, -1]], [[ 1, -1, -1],[ -1, -1, 0]], [[ 1, -1, -1],[ -1, 0, -1]], [[ -1, 1, -1],[ -1, -1, 0]], [[ -1, -1, 1],[ 0, -1, -1]], [[ -1, 1, 1],[ 0, -1, 1]], [[ 1, -1, 1],[ 1, 0, -1]], [[ 1, 1, -1],[ -1, 1, 0]], ],dtype=float), }, 'Bain': { 'cF' : np.array([ [[ 0, 1, 0],[ 1, 0, 0]], [[ 0, 0, 1],[ 0, 1, 0]], [[ 1, 0, 0],[ 0, 0, 1]], ],dtype=float), 'cI' : np.array([ [[ 0, 1, 1],[ 1, 0, 0]], [[ 1, 0, 1],[ 0, 1, 0]], [[ 1, 1, 0],[ 0, 0, 1]], ],dtype=float), }, 'Burgers' : { 'cI' : np.array([ [[ -1, 1, 1],[ 1, 1, 0]], [[ -1, 1, -1],[ 1, 1, 0]], [[ 1, 1, 1],[ 1, -1, 0]], [[ 1, 1, -1],[ 1, -1, 0]], [[ 1, 1, -1],[ 1, 0, 1]], [[ -1, 1, 1],[ 1, 0, 1]], [[ 1, 1, 1],[ -1, 0, 1]], [[ 1, -1, 1],[ -1, 0, 1]], [[ -1, 1, -1],[ 0, 1, 1]], [[ 1, 1, -1],[ 0, 1, 1]], [[ -1, 1, 1],[ 0, -1, 1]], [[ 1, 1, 1],[ 0, -1, 1]], ],dtype=float), 'hP' : np.array([ [[ -1, 2, -1, 0],[ 0, 0, 0, 1]], [[ -1, -1, 2, 0],[ 0, 0, 0, 1]], [[ -1, 2, -1, 0],[ 0, 0, 0, 1]], [[ -1, -1, 2, 0],[ 0, 0, 0, 1]], [[ -1, 2, -1, 0],[ 0, 0, 0, 1]], [[ -1, -1, 2, 0],[ 0, 0, 0, 1]], [[ -1, 2, -1, 0],[ 0, 0, 0, 1]], [[ -1, -1, 2, 0],[ 0, 0, 0, 1]], [[ -1, 2, -1, 0],[ 0, 0, 0, 1]], [[ -1, -1, 2, 0],[ 0, 0, 0, 1]], [[ -1, 2, -1, 0],[ 0, 0, 0, 1]], [[ -1, -1, 2, 0],[ 0, 0, 0, 1]], ],dtype=float), }, }