def test_from_voigt(self):
with self.assertRaises(ValueError):
TensorBase.from_voigt([[59.33, 28.08, 28.08, 0],
[28.08, 59.31, 28.07, 0],
[28.08, 28.07, 59.32, 0, 0],
[0, 0, 0, 26.35, 0], [0, 0, 0, 0, 26.35]])
# Rank 4
TensorBase.from_voigt([[59.33, 28.08, 28.08, 0, 0, 0],
[28.08, 59.31, 28.07, 0, 0, 0],
[28.08, 28.07, 59.32, 0, 0, 0],
[0, 0, 0, 26.35, 0, 0], [0, 0, 0, 0, 26.35, 0],
[0, 0, 0, 0, 0, 26.35]])
def test_from_voigt(self):
with self.assertRaises(ValueError):
TensorBase.from_voigt([[59.33, 28.08, 28.08, 0],
[28.08, 59.31, 28.07, 0],
[28.08, 28.07, 59.32, 0, 0],
[0, 0, 0, 26.35, 0],
[0, 0, 0, 0, 26.35]])
# Rank 4
TensorBase.from_voigt([[59.33, 28.08, 28.08, 0, 0, 0],
[28.08, 59.31, 28.07, 0, 0, 0],
[28.08, 28.07, 59.32, 0, 0, 0],
[0, 0, 0, 26.35, 0, 0],
[0, 0, 0, 0, 26.35, 0],
[0, 0, 0, 0, 0, 26.35]])
def toec_fit(strains, stresses, eq_stress = None, zero_crit=1e-10):
"""
A third-order elastic constant fitting function based on
central-difference derivatives with respect to distinct
strain states. The algorithm is summarized as follows:
1. Identify distinct strain states as sets of indices
for which nonzero strain values exist, typically
[(0), (1), (2), (3), (4), (5), (0, 1) etc.]
2. For each strain state, find and sort strains and
stresses by strain value.
3. Find first and second derivatives of each stress
with respect to scalar variable corresponding to
the smallest perturbation in the strain.
4. Use the pseudoinverse of a matrix-vector expression
corresponding to the parameterized stress-strain
relationship and multiply that matrix by the respective
calculated first or second derivatives from the
previous step.
5. Place the calculated second and third-order elastic
constants appropriately.
Args:
strains (nx3x3 array-like): Array of 3x3 strains
to use in fitting of TOEC and SOEC
stresses (nx3x3 array-like): Array of 3x3 stresses
to use in fitting of TOEC and SOEC. These
should be PK2 stresses.
eq_stress (3x3 array-like): stress corresponding to
equilibrium strain (i. e. "0" strain state).
If not specified, function will try to find
the state in the list of provided stresses
and strains. If not found, defaults to 0.
zero_crit (float): value for which strains below
are ignored in identifying strain states.
"""
if len(stresses) != len(strains):
raise ValueError("Length of strains and stresses are not equivalent")
vstresses = np.array([Stress(stress).voigt for stress in stresses])
vstrains = np.array([Strain(strain).voigt for strain in strains])
vstrains[np.abs(vstrains) < zero_crit] = 0
# Try to find eq_stress if not specified
if eq_stress is not None:
veq_stress = Stress(eq_stress).voigt
else:
veq_stress = vstresses[np.all(vstrains==0, axis=1)]
if veq_stress:
if np.shape(veq_stress) > 1 and not \
(abs(veq_stress - veq_stress[0]) < 1e-8).all():
raise ValueError("Multiple stresses found for equilibrium strain"
" state, please specify equilibrium stress or "
" remove extraneous stresses.")
veq_stress = veq_stress[0]
else:
veq_stress = np.zeros(6)
# Collect independent strain states:
independent = set([tuple(np.nonzero(vstrain)[0].tolist())
for vstrain in vstrains])
strain_states = []
dsde = np.zeros((6, len(independent)))
d2sde2 = np.zeros((6, len(independent)))
for n, ind in enumerate(independent):
# match strains with templates
template = np.zeros(6, dtype=bool)
np.put(template, ind, True)
template = np.tile(template, [vstresses.shape[0], 1])
mode = (template == (np.abs(vstrains) > 1e-10)).all(axis=1)
mstresses = vstresses[mode]
mstrains = vstrains[mode]
# add zero strain state
mstrains = np.vstack([mstrains, np.zeros(6)])
mstresses = np.vstack([mstresses, np.zeros(6)])
# sort strains/stresses by strain values
mstresses = mstresses[mstrains[:, ind[0]].argsort()]
mstrains = mstrains[mstrains[:, ind[0]].argsort()]
strain_states.append(mstrains[-1] / \
np.min(mstrains[-1][np.nonzero(mstrains[0])]))
diff = np.diff(mstrains, axis=0)
if not (abs(diff - diff[0]) < 1e-8).all():
raise ValueError("Stencil for strain state {} must be odd-sampling"
" centered at 0.".format(ind))
h = np.min(diff[np.nonzero(diff)])
coef1 = central_diff_weights(len(mstresses), 1)
coef2 = central_diff_weights(len(mstresses), 2)
if eq_stress is not None:
mstresses[len(mstresses) // 2] = veq_stress
dsde[:, n] = np.dot(np.transpose(mstresses), coef1) / h
d2sde2[:, n] = np.dot(np.transpose(mstresses), coef2) / h**2
m2i, m3i = generate_pseudo(strain_states)
s2vec = np.ravel(dsde.T)
c2vec = np.dot(m2i, s2vec)
c2 = np.zeros((6, 6))
c2[np.triu_indices(6)] = c2vec
c2 = c2 + c2.T - np.diag(np.diag(c2))
c3 = np.zeros((6, 6, 6))
s3vec = np.ravel(d2sde2.T)
c3vec = np.dot(m3i, s3vec)
list_indices = list(itertools.combinations_with_replacement(range(6), r=3))
indices_ij = itertools.combinations_with_replacement(range(6), r=3)
indices = list(itertools.combinations_with_replacement(range(6), r=3))
for n, (i, j, k) in enumerate(indices):
c3[i,j,k] = c3[i,k,j] = c3[j,i,k] = c3[j,k,i] = \
c3[k,i,j] = c3[k,j,i] = c3vec[n]
return TensorBase.from_voigt(c2), TensorBase.from_voigt(c3)
