def from_stress_dict(cls, stress_dict, tol=0.1, vasp=True):
"""
Constructs the elastic tensor from IndependentStrain-Stress dictionary
corresponding to legacy behavior of elasticity package.
Args:
stress_dict (dict): dictionary of stresses indexed by corresponding
IndependentStrain objects.
"""
inds = [(0, 0), (1, 1), (2, 2), (1, 2), (0, 2), (0, 1)]
c_ij = np.array([[
np.polyfit([
strain[ind1] for strain in list(stress_dict.keys())
if (strain.i, strain.j) == ind1
], [
stress_dict[strain][ind2]
for strain in list(stress_dict.keys())
if (strain.i, strain.j) == ind1
], 1)[0] for ind1 in inds
] for ind2 in inds])
if vasp:
c_ij *= -0.1 # Convert units/sign convention of vasp stress tensor
c_ij[0:,
3:] = 0.5 * c_ij[0:, 3:] # account for voigt doubling of e4,e5,e6
c_ij = SQTensor(c_ij)
c_ij = c_ij.zeroed(tol)
return cls(c_ij)
def __new__(cls, strain_matrix, dfm=None):
"""
Create a Strain object. Note that the constructor uses __new__
rather than __init__ according to the standard method of
subclassing numpy ndarrays. Note also that the default constructor
does not include the deformation gradient
Args:
strain_matrix (3x3 array-like): the 3x3 array-like
representing the Green-Lagrange strain
"""
obj = SQTensor(strain_matrix).view(cls)
obj._dfm = dfm
if not obj.is_symmetric():
raise ValueError("Strain objects must be initialized "
"with a symmetric array-like.")
if dfm is None:
warnings.warn("Constructing a strain object without a deformation "
"matrix makes many methods unusable. Use "
"Strain.from_deformation to construct a Strain object"
" from a deformation gradient.")
elif (np.array(dfm) - obj < 1e-5).all():
warnings.warn("Warning: deformation matrix does not correspond "
"to input strain_matrix value")
return obj
def __new__(cls, strain_matrix, dfm=None):
"""
Create a Strain object. Note that the constructor uses __new__
rather than __init__ according to the standard method of
subclassing numpy ndarrays. Note also that the default constructor
does not include the deformation gradient
Args:
strain_matrix (3x3 array-like): the 3x3 array-like
representing the Green-Lagrange strain
"""
obj = SQTensor(strain_matrix).view(cls)
obj._dfm = dfm
if not obj.is_symmetric():
raise ValueError("Strain objects must be initialized "
"with a symmetric array-like.")
if dfm is None:
warnings.warn(
"Constructing a strain object without a deformation "
"matrix makes many methods unusable. Use "
"Strain.from_deformation to construct a Strain object"
" from a deformation gradient.")
elif (np.array(dfm) - obj < 1e-5).all():
warnings.warn("Warning: deformation matrix does not correspond "
"to input strain_matrix value")
return obj
def test_zeroed(self):
self.assertArrayEqual(self.low_val.zeroed(),
SQTensor([[0, 1 + 1e-5, 0],
[1 + 1e-6, 0, 0],
[0, 0, 1 + 1e-5]]))
self.assertArrayEqual(self.low_val.zeroed(tol=1e-6),
SQTensor([[1e-6, 1 + 1e-5, 1e-6],
[1 + 1e-6, 1e-6, 1e-6],
[0, 0, 1 + 1e-5]]))
def test_rotate(self):
self.assertArrayAlmostEqual(self.non_symm.rotate(self.rotation),
SQTensor([[0.531, 0.485, 0.271],
[0.700, 0.5, 0.172],
[0.171, 0.233, 0.068]]),
decimal=3)
self.assertRaises(ValueError, self.non_symm.rotate, self.symm_sqtensor)
def test_properties(self):
# transpose
self.assertArrayEqual(self.non_symm.trans, SQTensor([[0.1, 0.4, 0.2],
[0.2, 0.5, 0.5],
[0.3, 0.6, 0.5]]))
self.assertArrayEqual(self.rand_sqtensor.trans,
np.transpose(self.rand_sqtensor))
self.assertArrayEqual(self.symm_sqtensor,
self.symm_sqtensor.trans)
# inverse
self.assertArrayEqual(self.non_symm.inv,
np.linalg.inv(self.non_symm))
with self.assertRaises(ValueError):
self.non_invertible.inv
# determinant
self.assertEqual(self.rand_sqtensor.det,
np.linalg.det(self.rand_sqtensor))
self.assertEqual(self.non_invertible.det,
0.0)
self.assertEqual(self.non_symm.det, 0.009)
# symmetrized
self.assertArrayEqual(self.rand_sqtensor.symmetrized,
0.5 * (self.rand_sqtensor + self.rand_sqtensor.trans))
self.assertArrayEqual(self.symm_sqtensor,
self.symm_sqtensor.symmetrized)
self.assertArrayAlmostEqual(self.non_symm.symmetrized,
SQTensor([[0.1, 0.3, 0.25],
[0.3, 0.5, 0.55],
[0.25, 0.55, 0.5]]))
# invariants
i1 = np.trace(self.rand_sqtensor)
i2 = self.rand_sqtensor[0, 0] * self.rand_sqtensor[1, 1] + \
self.rand_sqtensor[1, 1] * self.rand_sqtensor[2, 2] + \
self.rand_sqtensor[2, 2] * self.rand_sqtensor[0, 0] - \
self.rand_sqtensor[0, 1] * self.rand_sqtensor[1, 0] - \
self.rand_sqtensor[0, 2] * self.rand_sqtensor[2, 0] - \
self.rand_sqtensor[2, 1] * self.rand_sqtensor[1, 2]
i3 = np.linalg.det(self.rand_sqtensor)
self.assertArrayAlmostEqual([i1, i2, i3],
self.rand_sqtensor.principal_invariants)
def piola_kirchoff_1(self, def_grad):
"""
calculates the first Piola-Kirchoff stress
Args:
def_gradient (3x3 array-like): deformation gradient tensor
"""
if not self.is_symmetric:
raise ValueError("The stress tensor is not symmetric, \
PK stress is based on a symmetric stress tensor.")
def_grad = SQTensor(def_grad)
return def_grad.det*np.dot(self, def_grad.inv.trans)
def __new__(cls, stress_matrix):
"""
Create a Stress object. Note that the constructor uses __new__
rather than __init__ according to the standard method of
subclassing numpy ndarrays.
Args:
stress_matrix (3x3 array-like): the 3x3 array-like
representing the stress
"""
obj = SQTensor(stress_matrix).view(cls)
return obj
def __new__(cls, deformation_gradient):
"""
Create a Deformation object. Note that the constructor uses __new__
rather than __init__ according to the standard method of subclassing
numpy ndarrays.
Args:
deformation_gradient (3x3 array-like): the 3x3 array-like
representing the deformation gradient
"""
obj = SQTensor(deformation_gradient).view(cls)
return obj
def __new__(cls, input_matrix):
"""
Create an ElasticTensor object. The constructor throws an error if
the shape of the input_matrix argument is not 6x6, i. e. in Voigt-
notation. Also issues a warning if the input_matrix argument is
not symmetric. Note that the constructor uses __new__ rather than
__init__ according to the standard method of subclassing numpy
ndarrays.
Args:
input_matrix (6x6 array-like): the Voigt-notation 6x6 array-like
representing the elastic tensor
"""
obj = SQTensor(input_matrix).view(cls)
if obj.shape != (6, 6):
raise ValueError("Default elastic tensor constructor requires "
"input argument to be the Voigt-notation 6x6 "
"array. To construct from a 3x3x3x3 array, use "
"ElasticTensor.from_full_tensor")
if not obj.is_symmetric():
warnings.warn("Elastic tensor input is not symmetric!")
return obj
def __new__(cls, input_matrix):
"""
Create an ElasticTensor object. The constructor throws an error if
the shape of the input_matrix argument is not 6x6, i. e. in Voigt-
notation. Also issues a warning if the input_matrix argument is
not symmetric. Note that the constructor uses __new__ rather than
__init__ according to the standard method of subclassing numpy
ndarrays.
Args:
input_matrix (6x6 array-like): the Voigt-notation 6x6 array-like
representing the elastic tensor
"""
obj = SQTensor(input_matrix).view(cls)
if obj.shape != (6, 6):
raise ValueError("Default elastic tensor constructor requires "
"input argument to be the Voigt-notation 6x6 "
"array. To construct from a 3x3x3x3 array, use "
"ElasticTensor.from_full_tensor")
if not obj.is_symmetric():
warnings.warn("Elastic tensor input is not symmetric!")
return obj
def from_stress_dict(cls, stress_dict, tol=0.1, vasp=True):
"""
Constructs the elastic tensor from IndependentStrain-Stress dictionary
corresponding to legacy behavior of elasticity package.
Args:
stress_dict (dict): dictionary of stresses indexed by corresponding
IndependentStrain objects.
"""
inds = [(0, 0), (1, 1), (2, 2), (1, 2), (0, 2), (0, 1)]
c_ij = np.array([[np.polyfit([strain[ind1] for strain in list(stress_dict.keys())
if (strain.i, strain.j) == ind1],
[stress_dict[strain][ind2] for strain
in list(stress_dict.keys())
if (strain.i, strain.j) == ind1], 1)[0]
for ind1 in inds] for ind2 in inds])
if vasp:
c_ij *= -0.1 # Convert units/sign convention of vasp stress tensor
c_ij[0:, 3:] = 0.5 * c_ij[0:, 3:] # account for voigt doubling of e4,e5,e6
c_ij = SQTensor(c_ij)
c_ij = c_ij.zeroed(tol)
return cls(c_ij)
def piola_kirchoff_2(self, def_grad):
"""
calculates the second Piola-Kirchoff stress
Args:
f (3x3 array-like): rate of deformation tensor
"""
def_grad = SQTensor(def_grad)
if not self.is_symmetric:
raise ValueError("The stress tensor is not symmetric, \
PK stress is based on a symmetric stress tensor.")
return def_grad.det*np.dot(np.dot(def_grad.inv, self),
def_grad.inv.trans)
def setUp(self):
self.rand_sqtensor = SQTensor(np.random.randn(3, 3))
self.symm_sqtensor = SQTensor([[0.1, 0.3, 0.4],
[0.3, 0.5, 0.2],
[0.4, 0.2, 0.6]])
self.non_invertible = SQTensor([[0.1, 0, 0],
[0.2, 0, 0],
[0, 0, 0]])
self.non_symm = SQTensor([[0.1, 0.2, 0.3],
[0.4, 0.5, 0.6],
[0.2, 0.5, 0.5]])
self.low_val = SQTensor([[1e-6, 1 + 1e-5, 1e-6],
[1 + 1e-6, 1e-6, 1e-6],
[1e-7, 1e-7, 1 + 1e-5]])
self.low_val_2 = SQTensor([[1e-6, -1 - 1e-6, 1e-6],
[1 + 1e-7, 1e-6, 1e-6],
[1e-7, 1e-7, 1 + 1e-6]])
a = 3.14 * 42.5 / 180
self.rotation = SQTensor([[math.cos(a), 0, math.sin(a)],
[0, 1, 0],
[-math.sin(a), 0, math.cos(a)]])
def test_get_scaled(self):
self.assertArrayEqual(self.non_symm.get_scaled(10.),
SQTensor([[1, 2, 3], [4, 5, 6], [2, 5, 5]]))
class SQTensorTest(PymatgenTest):
def setUp(self):
self.rand_sqtensor = SQTensor(np.random.randn(3, 3))
self.symm_sqtensor = SQTensor([[0.1, 0.3, 0.4],
[0.3, 0.5, 0.2],
[0.4, 0.2, 0.6]])
self.non_invertible = SQTensor([[0.1, 0, 0],
[0.2, 0, 0],
[0, 0, 0]])
self.non_symm = SQTensor([[0.1, 0.2, 0.3],
[0.4, 0.5, 0.6],
[0.2, 0.5, 0.5]])
self.low_val = SQTensor([[1e-6, 1 + 1e-5, 1e-6],
[1 + 1e-6, 1e-6, 1e-6],
[1e-7, 1e-7, 1 + 1e-5]])
self.low_val_2 = SQTensor([[1e-6, -1 - 1e-6, 1e-6],
[1 + 1e-7, 1e-6, 1e-6],
[1e-7, 1e-7, 1 + 1e-6]])
a = 3.14 * 42.5 / 180
self.rotation = SQTensor([[math.cos(a), 0, math.sin(a)],
[0, 1, 0],
[-math.sin(a), 0, math.cos(a)]])
def test_new(self):
non_sq_matrix = [[0.1, 0.2, 0.1],
[0.1, 0.2, 0.3],
[0.1, 0.2, 0.3],
[0.1, 0.1, 0.1]]
bad_matrix = [[0.1, 0.2],
[0.2, 0.3, 0.4],
[0.2, 0.3, 0.5]]
self.assertRaises(ValueError, SQTensor, non_sq_matrix)
self.assertRaises(ValueError, SQTensor, bad_matrix)
def test_properties(self):
# transpose
self.assertArrayEqual(self.non_symm.trans, SQTensor([[0.1, 0.4, 0.2],
[0.2, 0.5, 0.5],
[0.3, 0.6, 0.5]]))
self.assertArrayEqual(self.rand_sqtensor.trans,
np.transpose(self.rand_sqtensor))
self.assertArrayEqual(self.symm_sqtensor,
self.symm_sqtensor.trans)
# inverse
self.assertArrayEqual(self.non_symm.inv,
np.linalg.inv(self.non_symm))
with self.assertRaises(ValueError):
self.non_invertible.inv
# determinant
self.assertEqual(self.rand_sqtensor.det,
np.linalg.det(self.rand_sqtensor))
self.assertEqual(self.non_invertible.det,
0.0)
self.assertEqual(self.non_symm.det, 0.009)
# symmetrized
self.assertArrayEqual(self.rand_sqtensor.symmetrized,
0.5 * (self.rand_sqtensor + self.rand_sqtensor.trans))
self.assertArrayEqual(self.symm_sqtensor,
self.symm_sqtensor.symmetrized)
self.assertArrayAlmostEqual(self.non_symm.symmetrized,
SQTensor([[0.1, 0.3, 0.25],
[0.3, 0.5, 0.55],
[0.25, 0.55, 0.5]]))
# invariants
i1 = np.trace(self.rand_sqtensor)
i2 = self.rand_sqtensor[0, 0] * self.rand_sqtensor[1, 1] + \
self.rand_sqtensor[1, 1] * self.rand_sqtensor[2, 2] + \
self.rand_sqtensor[2, 2] * self.rand_sqtensor[0, 0] - \
self.rand_sqtensor[0, 1] * self.rand_sqtensor[1, 0] - \
self.rand_sqtensor[0, 2] * self.rand_sqtensor[2, 0] - \
self.rand_sqtensor[2, 1] * self.rand_sqtensor[1, 2]
i3 = np.linalg.det(self.rand_sqtensor)
self.assertArrayAlmostEqual([i1, i2, i3],
self.rand_sqtensor.principal_invariants)
def test_symmetry(self):
self.assertTrue(self.symm_sqtensor.is_symmetric())
self.assertFalse(self.non_symm.is_symmetric())
self.assertTrue(self.low_val.is_symmetric())
self.assertFalse(self.low_val.is_symmetric(tol=1e-8))
def test_rotation(self):
self.assertTrue(self.rotation.is_rotation())
self.assertFalse(self.symm_sqtensor.is_rotation())
self.assertTrue(self.low_val_2.is_rotation())
self.assertFalse(self.low_val_2.is_rotation(tol=1e-8))
def test_rotate(self):
self.assertArrayAlmostEqual(self.non_symm.rotate(self.rotation),
SQTensor([[0.531, 0.485, 0.271],
[0.700, 0.5, 0.172],
[0.171, 0.233, 0.068]]),
decimal=3)
self.assertRaises(ValueError, self.non_symm.rotate, self.symm_sqtensor)
def test_get_scaled(self):
self.assertArrayEqual(self.non_symm.get_scaled(10.),
SQTensor([[1, 2, 3], [4, 5, 6], [2, 5, 5]]))
def test_polar_decomposition(self):
u, p = self.rand_sqtensor.polar_decomposition()
self.assertArrayAlmostEqual(np.dot(u, p), self.rand_sqtensor)
self.assertArrayAlmostEqual(np.eye(3),
np.dot(u, np.conjugate(np.transpose(u))))
def test_zeroed(self):
self.assertArrayEqual(self.low_val.zeroed(),
SQTensor([[0, 1 + 1e-5, 0],
[1 + 1e-6, 0, 0],
[0, 0, 1 + 1e-5]]))
self.assertArrayEqual(self.low_val.zeroed(tol=1e-6),
SQTensor([[1e-6, 1 + 1e-5, 1e-6],
[1 + 1e-6, 1e-6, 1e-6],
[0, 0, 1 + 1e-5]]))
def setUp(self):
self.rand_sqtensor = SQTensor(np.random.randn(3, 3))
self.symm_sqtensor = SQTensor([[0.1, 0.3, 0.4], [0.3, 0.5, 0.2],
[0.4, 0.2, 0.6]])
self.non_invertible = SQTensor([[0.1, 0, 0], [0.2, 0, 0], [0, 0, 0]])
self.non_symm = SQTensor([[0.1, 0.2, 0.3], [0.4, 0.5, 0.6],
[0.2, 0.5, 0.5]])
self.low_val = SQTensor([[1e-6, 1 + 1e-5,
1e-6], [1 + 1e-6, 1e-6, 1e-6],
[1e-7, 1e-7, 1 + 1e-5]])
self.low_val_2 = SQTensor([[1e-6, -1 - 1e-6, 1e-6],
[1 + 1e-7, 1e-6, 1e-6],
[1e-7, 1e-7, 1 + 1e-6]])
a = 3.14 * 42.5 / 180
self.rotation = SQTensor([[math.cos(a), 0, math.sin(a)], [0, 1, 0],
[-math.sin(a), 0,
math.cos(a)]])
class SQTensorTest(PymatgenTest):
def setUp(self):
self.rand_sqtensor = SQTensor(np.random.randn(3, 3))
self.symm_sqtensor = SQTensor([[0.1, 0.3, 0.4], [0.3, 0.5, 0.2],
[0.4, 0.2, 0.6]])
self.non_invertible = SQTensor([[0.1, 0, 0], [0.2, 0, 0], [0, 0, 0]])
self.non_symm = SQTensor([[0.1, 0.2, 0.3], [0.4, 0.5, 0.6],
[0.2, 0.5, 0.5]])
self.low_val = SQTensor([[1e-6, 1 + 1e-5,
1e-6], [1 + 1e-6, 1e-6, 1e-6],
[1e-7, 1e-7, 1 + 1e-5]])
self.low_val_2 = SQTensor([[1e-6, -1 - 1e-6, 1e-6],
[1 + 1e-7, 1e-6, 1e-6],
[1e-7, 1e-7, 1 + 1e-6]])
a = 3.14 * 42.5 / 180
self.rotation = SQTensor([[math.cos(a), 0, math.sin(a)], [0, 1, 0],
[-math.sin(a), 0,
math.cos(a)]])
def test_new(self):
non_sq_matrix = [[0.1, 0.2, 0.1], [0.1, 0.2, 0.3], [0.1, 0.2, 0.3],
[0.1, 0.1, 0.1]]
bad_matrix = [[0.1, 0.2], [0.2, 0.3, 0.4], [0.2, 0.3, 0.5]]
self.assertRaises(ValueError, SQTensor, non_sq_matrix)
self.assertRaises(ValueError, SQTensor, bad_matrix)
def test_properties(self):
# transpose
self.assertArrayEqual(
self.non_symm.trans,
SQTensor([[0.1, 0.4, 0.2], [0.2, 0.5, 0.5], [0.3, 0.6, 0.5]]))
self.assertArrayEqual(self.rand_sqtensor.trans,
np.transpose(self.rand_sqtensor))
self.assertArrayEqual(self.symm_sqtensor, self.symm_sqtensor.trans)
# inverse
self.assertArrayEqual(self.non_symm.inv, np.linalg.inv(self.non_symm))
with self.assertRaises(ValueError):
self.non_invertible.inv
# determinant
self.assertEqual(self.rand_sqtensor.det,
np.linalg.det(self.rand_sqtensor))
self.assertEqual(self.non_invertible.det, 0.0)
self.assertEqual(self.non_symm.det, 0.009)
# symmetrized
self.assertArrayEqual(
self.rand_sqtensor.symmetrized,
0.5 * (self.rand_sqtensor + self.rand_sqtensor.trans))
self.assertArrayEqual(self.symm_sqtensor,
self.symm_sqtensor.symmetrized)
self.assertArrayAlmostEqual(
self.non_symm.symmetrized,
SQTensor([[0.1, 0.3, 0.25], [0.3, 0.5, 0.55], [0.25, 0.55, 0.5]]))
# invariants
i1 = np.trace(self.rand_sqtensor)
i2 = self.rand_sqtensor[0, 0] * self.rand_sqtensor[1, 1] + \
self.rand_sqtensor[1, 1] * self.rand_sqtensor[2, 2] + \
self.rand_sqtensor[2, 2] * self.rand_sqtensor[0, 0] - \
self.rand_sqtensor[0, 1] * self.rand_sqtensor[1, 0] - \
self.rand_sqtensor[0, 2] * self.rand_sqtensor[2, 0] - \
self.rand_sqtensor[2, 1] * self.rand_sqtensor[1, 2]
i3 = np.linalg.det(self.rand_sqtensor)
self.assertArrayAlmostEqual([i1, i2, i3],
self.rand_sqtensor.principal_invariants)
def test_symmetry(self):
self.assertTrue(self.symm_sqtensor.is_symmetric())
self.assertFalse(self.non_symm.is_symmetric())
self.assertTrue(self.low_val.is_symmetric())
self.assertFalse(self.low_val.is_symmetric(tol=1e-8))
def test_rotation(self):
self.assertTrue(self.rotation.is_rotation())
self.assertFalse(self.symm_sqtensor.is_rotation())
self.assertTrue(self.low_val_2.is_rotation())
self.assertFalse(self.low_val_2.is_rotation(tol=1e-8))
def test_rotate(self):
self.assertArrayAlmostEqual(self.non_symm.rotate(self.rotation),
SQTensor([[0.531, 0.485, 0.271],
[0.700, 0.5, 0.172],
[0.171, 0.233, 0.068]]),
decimal=3)
self.assertRaises(ValueError, self.non_symm.rotate, self.symm_sqtensor)
def test_get_scaled(self):
self.assertArrayEqual(self.non_symm.get_scaled(10.),
SQTensor([[1, 2, 3], [4, 5, 6], [2, 5, 5]]))
def test_polar_decomposition(self):
u, p = self.rand_sqtensor.polar_decomposition()
self.assertArrayAlmostEqual(np.dot(u, p), self.rand_sqtensor)
self.assertArrayAlmostEqual(np.eye(3),
np.dot(u, np.conjugate(np.transpose(u))))
def test_zeroed(self):
self.assertArrayEqual(
self.low_val.zeroed(),
SQTensor([[0, 1 + 1e-5, 0], [1 + 1e-6, 0, 0], [0, 0, 1 + 1e-5]]))
self.assertArrayEqual(
self.low_val.zeroed(tol=1e-6),
SQTensor([[1e-6, 1 + 1e-5, 1e-6], [1 + 1e-6, 1e-6, 1e-6],
[0, 0, 1 + 1e-5]]))
self.assertArrayEqual(
SQTensor([[1e-6, -30, 1], [1e-7, 1, 0], [1e-8, 0, 1]]).zeroed(),
SQTensor([[0, -30, 1], [0, 1, 0], [0, 0, 1]]))
