def calculate(dbf,
comps,
phases,
mode=None,
output='GM',
fake_points=False,
**kwargs):
"""
Sample the property surface of 'output' containing the specified
components and phases. Model parameters are taken from 'dbf' and any
state variables (T, P, etc.) can be specified as keyword arguments.
Parameters
----------
dbf : Database
Thermodynamic database containing the relevant parameters.
comps : str or sequence
Names of components to consider in the calculation.
phases : str or sequence
Names of phases to consider in the calculation.
mode : string, optional
See 'make_callable' docstring for details.
output : string, optional
Model attribute to sample.
fake_points : bool, optional (Default: False)
If True, the first few points of the output surface will be fictitious
points used to define an equilibrium hyperplane guaranteed to be above
all the other points. This is used for convex hull computations.
points : ndarray or a dict of phase names to ndarray, optional
Columns of ndarrays must be internal degrees of freedom (site fractions), sorted.
If this is not specified, points will be generated automatically.
pdens : int, a dict of phase names to int, or a seq of both, optional
Number of points to sample per degree of freedom.
model : Model, a dict of phase names to Model, or a seq of both, optional
Model class to use for each phase.
Returns
-------
xray.Dataset of the sampled attribute as a function of state variables
Examples
--------
None yet.
"""
# Here we check for any keyword arguments that are special, i.e.,
# there may be keyword arguments that aren't state variables
pdens_dict = unpack_kwarg(kwargs.pop('pdens', 2000), default_arg=2000)
points_dict = unpack_kwarg(kwargs.pop('points', None), default_arg=None)
model_dict = unpack_kwarg(kwargs.pop('model', Model), default_arg=Model)
callable_dict = unpack_kwarg(kwargs.pop('callables', None),
default_arg=None)
if isinstance(phases, str):
phases = [phases]
if isinstance(comps, str):
comps = [comps]
components = [x for x in sorted(comps) if not x.startswith('VA')]
# Convert keyword strings to proper state variable objects
# If we don't do this, sympy will get confused during substitution
statevar_dict = collections.OrderedDict((v.StateVariable(key), unpack_condition(value)) \
for (key, value) in sorted(kwargs.items()))
str_statevar_dict = collections.OrderedDict((str(key), unpack_condition(value)) \
for (key, value) in statevar_dict.items())
all_phase_data = []
comp_sets = {}
largest_energy = -np.inf
maximum_internal_dof = 0
# Consider only the active phases
active_phases = dict((name.upper(), dbf.phases[name.upper()]) \
for name in unpack_phases(phases))
for phase_name, phase_obj in sorted(active_phases.items()):
# Build the symbolic representation of the energy
mod = model_dict[phase_name]
# if this is an object type, we need to construct it
if isinstance(mod, type):
try:
model_dict[phase_name] = mod = mod(dbf, comps, phase_name)
except DofError:
# we can't build the specified phase because the
# specified components aren't found in every sublattice
# we'll just skip it
logger.warning(
"""Suspending specified phase %s due to
some sublattices containing only unspecified components""",
phase_name)
continue
if points_dict[phase_name] is None:
try:
out = getattr(mod, output)
maximum_internal_dof = max(maximum_internal_dof,
len(out.atoms(v.SiteFraction)))
except AttributeError:
raise AttributeError(
'Missing Model attribute {0} specified for {1}'.format(
output, mod.__class__))
else:
maximum_internal_dof = max(
maximum_internal_dof,
np.asarray(points_dict[phase_name]).shape[-1])
for phase_name, phase_obj in sorted(active_phases.items()):
try:
mod = model_dict[phase_name]
except KeyError:
continue
# Construct an ordered list of the variables
variables, sublattice_dof = generate_dof(phase_obj, mod.components)
# Build the "fast" representation of that model
if callable_dict[phase_name] is None:
out = getattr(mod, output)
# As a last resort, treat undefined symbols as zero
# But warn the user when we do this
# This is consistent with TC's behavior
undefs = list(out.atoms(Symbol) - out.atoms(v.StateVariable))
for undef in undefs:
out = out.xreplace({undef: float(0)})
logger.warning(
'Setting undefined symbol %s for phase %s to zero', undef,
phase_name)
comp_sets[phase_name] = make_callable(out, \
list(statevar_dict.keys()) + variables, mode=mode)
else:
comp_sets[phase_name] = callable_dict[phase_name]
points = points_dict[phase_name]
if points is None:
# Eliminate pure vacancy endmembers from the calculation
vacancy_indices = list()
for idx, sublattice in enumerate(phase_obj.constituents):
active_in_subl = sorted(
set(phase_obj.constituents[idx]).intersection(comps))
if 'VA' in active_in_subl and 'VA' in sorted(comps):
vacancy_indices.append(active_in_subl.index('VA'))
if len(vacancy_indices) != len(phase_obj.constituents):
vacancy_indices = None
logger.debug('vacancy_indices: %s', vacancy_indices)
# Add all endmembers to guarantee their presence
points = endmember_matrix(sublattice_dof,
vacancy_indices=vacancy_indices)
# Sample composition space for more points
if sum(sublattice_dof) > len(sublattice_dof):
points = np.concatenate(
(points,
point_sample(sublattice_dof,
pdof=pdens_dict[phase_name])))
# If there are nontrivial sublattices with vacancies in them,
# generate a set of points where their fraction is zero and renormalize
for idx, sublattice in enumerate(phase_obj.constituents):
if 'VA' in set(sublattice) and len(sublattice) > 1:
var_idx = variables.index(
v.SiteFraction(phase_name, idx, 'VA'))
addtl_pts = np.copy(points)
# set vacancy fraction to log-spaced between 1e-10 and 1e-6
addtl_pts[:, var_idx] = np.power(
10.0, -10.0 * (1.0 - addtl_pts[:, var_idx]))
# renormalize site fractions
cur_idx = 0
for ctx in sublattice_dof:
end_idx = cur_idx + ctx
addtl_pts[:, cur_idx:end_idx] /= \
addtl_pts[:, cur_idx:end_idx].sum(axis=1)[:, None]
cur_idx = end_idx
# add to points matrix
points = np.concatenate((points, addtl_pts), axis=0)
# Filter out nan's that may have slipped in if we sampled too high a vacancy concentration
# Issues with this appear to be platform-dependent
points = points[~np.isnan(points).any(axis=-1)]
# Ensure that points has the correct dimensions and dtype
points = np.atleast_2d(np.asarray(points, dtype=np.float))
phase_ds = _compute_phase_values(phase_obj, components, variables,
str_statevar_dict, points,
comp_sets[phase_name], output,
maximum_internal_dof)
# largest_energy is really only relevant if fake_points is set
if fake_points:
largest_energy = max(phase_ds[output].max(), largest_energy)
all_phase_data.append(phase_ds)
if fake_points:
if output != 'GM':
raise ValueError(
'fake_points=True should only be used with output=\'GM\'')
phase_ds = _generate_fake_points(components, statevar_dict,
largest_energy, output,
maximum_internal_dof)
final_ds = xray.concat(itertools.chain([phase_ds], all_phase_data),
dim='points')
else:
# speedup for single-phase case (found by profiling)
if len(all_phase_data) > 1:
final_ds = xray.concat(all_phase_data, dim='points')
else:
final_ds = all_phase_data[0]
if (not fake_points) and (len(all_phase_data) == 1):
pass
else:
# Reset the points dimension to use a single global index
final_ds['points'] = np.arange(len(final_ds.points))
return final_ds
def equilibrium(dbf, comps, phases, conditions, **kwargs):
"""
Calculate the equilibrium state of a system containing the specified
components and phases, under the specified conditions.
Model parameters are taken from 'dbf'.
Parameters
----------
dbf : Database
Thermodynamic database containing the relevant parameters.
comps : list
Names of components to consider in the calculation.
phases : list or dict
Names of phases to consider in the calculation.
conditions : dict or (list of dict)
StateVariables and their corresponding value.
verbose : bool, optional (Default: True)
Show progress of calculations.
grid_opts : dict, optional
Keyword arguments to pass to the initial grid routine.
Returns
-------
Structured equilibrium calculation.
Examples
--------
None yet.
"""
active_phases = unpack_phases(phases) or sorted(dbf.phases.keys())
comps = sorted(comps)
indep_vars = ['T', 'P']
grid_opts = kwargs.pop('grid_opts', dict())
verbose = kwargs.pop('verbose', True)
phase_records = dict()
callable_dict = kwargs.pop('callables', dict())
grad_callable_dict = kwargs.pop('grad_callables', dict())
hess_callable_dict = kwargs.pop('hess_callables', dict())
points_dict = dict()
maximum_internal_dof = 0
conds = OrderedDict((key, unpack_condition(value))
for key, value in sorted(conditions.items(), key=str))
str_conds = OrderedDict((str(key), value) for key, value in conds.items())
indep_vals = list([float(x) for x in np.atleast_1d(val)]
for key, val in str_conds.items() if key in indep_vars)
components = [x for x in sorted(comps) if not x.startswith('VA')]
# Construct models for each phase; prioritize user models
models = unpack_kwarg(kwargs.pop('model', Model), default_arg=Model)
if verbose:
print('Components:', ' '.join(comps))
print('Phases:', end=' ')
for name in active_phases:
mod = models[name]
if isinstance(mod, type):
models[name] = mod = mod(dbf, comps, name)
variables = sorted(mod.energy.atoms(v.StateVariable).union(
{key
for key in conditions.keys() if key in [v.T, v.P]}),
key=str)
site_fracs = sorted(mod.energy.atoms(v.SiteFraction), key=str)
maximum_internal_dof = max(maximum_internal_dof, len(site_fracs))
# Extra factor '1e-100...' is to work around an annoying broadcasting bug for zero gradient entries
#models[name].models['_broadcaster'] = 1e-100 * Mul(*variables) ** 3
out = models[name].energy
undefs = list(out.atoms(Symbol) - out.atoms(v.StateVariable))
for undef in undefs:
out = out.xreplace({undef: float(0)})
callable_dict[name], grad_callable_dict[name], hess_callable_dict[name] = \
build_functions(out, [v.P, v.T] + site_fracs)
# Adjust gradient by the approximate chemical potentials
hyperplane = Add(*[
v.MU(i) * mole_fraction(dbf.phases[name], comps, i) for i in comps
if i != 'VA'
])
plane_obj, plane_grad, plane_hess = build_functions(
hyperplane, [v.MU(i) for i in comps if i != 'VA'] + site_fracs)
phase_records[name.upper()] = PhaseRecord(
variables=variables,
grad=grad_callable_dict[name],
hess=hess_callable_dict[name],
plane_grad=plane_grad,
plane_hess=plane_hess)
if verbose:
print(name, end=' ')
if verbose:
print('[done]', end='\n')
# 'calculate' accepts conditions through its keyword arguments
grid_opts.update(
{key: value
for key, value in str_conds.items() if key in indep_vars})
if 'pdens' not in grid_opts:
grid_opts['pdens'] = 100
coord_dict = str_conds.copy()
coord_dict['vertex'] = np.arange(len(components))
grid_shape = np.meshgrid(*coord_dict.values(), indexing='ij',
sparse=False)[0].shape
coord_dict['component'] = components
if verbose:
print('Computing initial grid', end=' ')
grid = calculate(dbf,
comps,
active_phases,
output='GM',
model=models,
callables=callable_dict,
fake_points=True,
**grid_opts)
if verbose:
print('[{0} points, {1}]'.format(len(grid.points),
sizeof_fmt(grid.nbytes)),
end='\n')
properties = xray.Dataset(
{
'NP': (list(str_conds.keys()) + ['vertex'], np.empty(grid_shape)),
'GM': (list(str_conds.keys()), np.empty(grid_shape[:-1])),
'MU':
(list(str_conds.keys()) + ['component'], np.empty(grid_shape)),
'points': (list(str_conds.keys()) + ['vertex'],
np.empty(grid_shape, dtype=np.int))
},
coords=coord_dict,
attrs={'iterations': 1},
)
# Store the potentials from the previous iteration
current_potentials = properties.MU.copy()
for iteration in range(MAX_ITERATIONS):
if verbose:
print('Computing convex hull [iteration {}]'.format(
properties.attrs['iterations']))
# lower_convex_hull will modify properties
lower_convex_hull(grid, properties)
progress = np.abs(current_potentials - properties.MU).values
converged = (progress < MIN_PROGRESS).all(axis=-1)
if verbose:
print('progress', progress.max(),
'[{} conditions updated]'.format(np.sum(~converged)))
if progress.max() < MIN_PROGRESS:
if verbose:
print('Convergence achieved')
break
current_potentials[...] = properties.MU.values
if verbose:
print('Refining convex hull')
# Insert extra dimensions for non-T,P conditions so GM broadcasts correctly
energy_broadcast_shape = grid.GM.values.shape[:len(indep_vals)] + \
(1,) * (len(str_conds) - len(indep_vals)) + (grid.GM.values.shape[-1],)
driving_forces = np.einsum('...i,...i',
properties.MU.values[..., np.newaxis, :].astype(np.float),
grid.X.values[np.index_exp[...] +
(np.newaxis,) * (len(str_conds) - len(indep_vals)) +
np.index_exp[:, :]].astype(np.float)) - \
grid.GM.values.view().reshape(energy_broadcast_shape)
for name in active_phases:
dof = len(models[name].energy.atoms(v.SiteFraction))
current_phase_indices = (grid.Phase.values == name
).reshape(energy_broadcast_shape[:-1] +
(-1, ))
# Broadcast to capture all conditions
current_phase_indices = np.broadcast_arrays(
current_phase_indices, np.empty(driving_forces.shape))[0]
# This reshape is safe as long as phases have the same number of points at all indep. conditions
current_phase_driving_forces = driving_forces[
current_phase_indices].reshape(
current_phase_indices.shape[:-1] + (-1, ))
# Note: This works as long as all points are in the same phase order for all T, P
current_site_fractions = grid.Y.values[..., current_phase_indices[
(0, ) * len(str_conds)], :]
if np.sum(
current_site_fractions[(0, ) *
len(indep_vals)][..., :dof]) == dof:
# All site fractions are 1, aka zero internal degrees of freedom
# Impossible to refine these points, so skip this phase
points_dict[name] = current_site_fractions[
(0, ) * len(indep_vals)][..., :dof]
continue
# Find the N points with largest driving force for a given set of conditions
# Remember that driving force has a sign, so we want the "most positive" values
# N is the number of components, in this context
# N points define a 'best simplex' for every set of conditions
# We also need to restrict ourselves to one phase at a time
trial_indices = np.argpartition(current_phase_driving_forces,
-len(components),
axis=-1)[..., -len(components):]
trial_indices = trial_indices.ravel()
statevar_indices = np.unravel_index(
np.arange(
np.multiply.reduce(properties.GM.values.shape +
(len(components), ))),
properties.GM.values.shape +
(len(components), ))[:len(indep_vals)]
points = current_site_fractions[np.index_exp[statevar_indices +
(trial_indices, )]]
points.shape = properties.points.shape[:-1] + (
-1, maximum_internal_dof)
# The Y arrays have been padded, so we should slice off the padding
points = points[..., :dof]
#print('Starting points shape: ', points.shape)
#print(points)
if len(points) == 0:
if name in points_dict:
del points_dict[name]
# No nearly stable points: skip this phase
continue
num_vars = len(phase_records[name].variables)
plane_grad = phase_records[name].plane_grad
plane_hess = phase_records[name].plane_hess
statevar_grid = np.meshgrid(*itertools.chain(indep_vals),
sparse=True,
indexing='ij')
# TODO: A more sophisticated treatment of constraints
num_constraints = len(dbf.phases[name].sublattices)
constraint_jac = np.zeros(
(num_constraints, num_vars - len(indep_vars)))
# Independent variables are always fixed (in this limited implementation)
#for idx in range(len(indep_vals)):
# constraint_jac[idx, idx] = 1
# This is for site fraction balance constraints
var_idx = 0 #len(indep_vals)
for idx in range(len(dbf.phases[name].sublattices)):
active_in_subl = set(
dbf.phases[name].constituents[idx]).intersection(comps)
constraint_jac[idx, var_idx:var_idx + len(active_in_subl)] = 1
var_idx += len(active_in_subl)
newton_iteration = 0
while newton_iteration < MAX_NEWTON_ITERATIONS:
flattened_points = points.reshape(
points.shape[:len(indep_vals)] + (-1, points.shape[-1]))
grad_args = itertools.chain(
[i[..., None] for i in statevar_grid], [
flattened_points[..., i]
for i in range(flattened_points.shape[-1])
])
grad = np.array(phase_records[name].grad(*grad_args),
dtype=np.float)
# Remove derivatives wrt T,P
grad = grad[..., len(indep_vars):]
grad.shape = points.shape
grad[np.isnan(grad).any(
axis=-1
)] = 0 # This is necessary for gradients on the edge of space
hess_args = itertools.chain(
[i[..., None] for i in statevar_grid], [
flattened_points[..., i]
for i in range(flattened_points.shape[-1])
])
hess = np.array(phase_records[name].hess(*hess_args),
dtype=np.float)
# Remove derivatives wrt T,P
hess = hess[..., len(indep_vars):, len(indep_vars):]
hess.shape = points.shape + (hess.shape[-1], )
hess[np.isnan(hess).any(axis=(-2,
-1))] = np.eye(hess.shape[-1])
plane_args = itertools.chain([
properties.MU.values[..., i][..., None]
for i in range(properties.MU.shape[-1])
], [points[..., i] for i in range(points.shape[-1])])
cast_grad = np.array(plane_grad(*plane_args), dtype=np.float)
# Remove derivatives wrt chemical potentials
cast_grad = cast_grad[..., properties.MU.shape[-1]:]
grad = grad - cast_grad
plane_args = itertools.chain([
properties.MU.values[..., i][..., None]
for i in range(properties.MU.shape[-1])
], [points[..., i] for i in range(points.shape[-1])])
cast_hess = np.array(plane_hess(*plane_args), dtype=np.float)
# Remove derivatives wrt chemical potentials
cast_hess = cast_hess[..., properties.MU.shape[-1]:,
properties.MU.shape[-1]:]
cast_hess = -cast_hess + hess
hess = cast_hess.astype(np.float, copy=False)
try:
e_matrix = np.linalg.inv(hess)
except np.linalg.LinAlgError:
print(hess)
raise
current = calculate(
dbf,
comps,
name,
output='GM',
model=models,
callables=callable_dict,
fake_points=False,
points=points.reshape(points.shape[:len(indep_vals)] +
(-1, points.shape[-1])),
**grid_opts)
current_plane = np.multiply(
current.X.values.reshape(points.shape[:-1] +
(len(components), )),
properties.MU.values[..., np.newaxis, :]).sum(axis=-1)
current_df = current.GM.values.reshape(
points.shape[:-1]) - current_plane
#print('Inv hess check: ', np.isnan(e_matrix).any())
#print('grad check: ', np.isnan(grad).any())
dy_unconstrained = -np.einsum('...ij,...j->...i', e_matrix,
grad)
#print('dy_unconstrained check: ', np.isnan(dy_unconstrained).any())
proj_matrix = np.dot(e_matrix, constraint_jac.T)
inv_matrix = np.rollaxis(np.dot(constraint_jac, proj_matrix),
0, -1)
inv_term = np.linalg.inv(inv_matrix)
#print('inv_term check: ', np.isnan(inv_term).any())
first_term = np.einsum('...ij,...jk->...ik', proj_matrix,
inv_term)
#print('first_term check: ', np.isnan(first_term).any())
# Normally a term for the residual here
# We only choose starting points which obey the constraints, so r = 0
cons_summation = np.einsum('...i,...ji->...j',
dy_unconstrained, constraint_jac)
#print('cons_summation check: ', np.isnan(cons_summation).any())
cons_correction = np.einsum('...ij,...j->...i', first_term,
cons_summation)
#print('cons_correction check: ', np.isnan(cons_correction).any())
dy_constrained = dy_unconstrained - cons_correction
#print('dy_constrained check: ', np.isnan(dy_constrained).any())
# TODO: Support for adaptive changing independent variable steps
new_direction = dy_constrained
#print('new_direction', new_direction)
#print('points', points)
# Backtracking line search
if np.isnan(new_direction).any():
print('new_direction', new_direction)
#print('Convergence angle:', -(grad*new_direction).sum(axis=-1) / (np.linalg.norm(grad, axis=-1) * np.linalg.norm(new_direction, axis=-1)))
new_points = points + INITIAL_STEP_SIZE * new_direction
alpha = np.full(new_points.shape[:-1],
INITIAL_STEP_SIZE,
dtype=np.float)
alpha[np.all(np.linalg.norm(new_direction, axis=-1) <
MIN_DIRECTION_NORM,
axis=-1)] = 0
negative_points = np.any(new_points < 0., axis=-1)
while np.any(negative_points):
alpha[negative_points] *= 0.5
new_points = points + alpha[...,
np.newaxis] * new_direction
negative_points = np.any(new_points < 0., axis=-1)
# Backtracking line search
# alpha now contains maximum possible values that keep us inside the space
# but we don't just want to take the biggest step; we want the biggest step which reduces energy
new_points = new_points.reshape(
new_points.shape[:len(indep_vals)] +
(-1, new_points.shape[-1]))
candidates = calculate(dbf,
comps,
name,
output='GM',
model=models,
callables=callable_dict,
fake_points=False,
points=new_points,
**grid_opts)
candidate_plane = np.multiply(
candidates.X.values.reshape(points.shape[:-1] +
(len(components), )),
properties.MU.values[..., np.newaxis, :]).sum(axis=-1)
energy_diff = (candidates.GM.values.reshape(
new_direction.shape[:-1]) - candidate_plane) - current_df
new_points.shape = new_direction.shape
bad_steps = energy_diff > alpha * 1e-4 * (new_direction *
grad).sum(axis=-1)
backtracking_iterations = 0
while np.any(bad_steps):
alpha[bad_steps] *= 0.5
new_points = points + alpha[...,
np.newaxis] * new_direction
#print('new_points', new_points)
#print('bad_steps', bad_steps)
new_points = new_points.reshape(
new_points.shape[:len(indep_vals)] +
(-1, new_points.shape[-1]))
candidates = calculate(dbf,
comps,
name,
output='GM',
model=models,
callables=callable_dict,
fake_points=False,
points=new_points,
**grid_opts)
candidate_plane = np.multiply(
candidates.X.values.reshape(points.shape[:-1] +
(len(components), )),
properties.MU.values[..., np.newaxis, :]).sum(axis=-1)
energy_diff = (candidates.GM.values.reshape(
new_direction.shape[:-1]) -
candidate_plane) - current_df
#print('energy_diff', energy_diff)
new_points.shape = new_direction.shape
bad_steps = energy_diff > alpha * 1e-4 * (
new_direction * grad).sum(axis=-1)
backtracking_iterations += 1
if backtracking_iterations > MAX_BACKTRACKING:
break
biggest_step = np.max(
np.linalg.norm(new_points - points, axis=-1))
if biggest_step < 1e-2:
if verbose:
print('N-R convergence on mini-iteration',
newton_iteration, '[{}]'.format(name))
points = new_points
break
if verbose:
#print('Biggest step:', biggest_step)
#print('points', points)
#print('grad of points', grad)
#print('new_direction', new_direction)
#print('alpha', alpha)
#print('new_points', new_points)
pass
points = new_points
newton_iteration += 1
new_points = points.reshape(points.shape[:len(indep_vals)] +
(-1, points.shape[-1]))
new_points = np.concatenate(
(current_site_fractions[..., :dof], new_points), axis=-2)
points_dict[name] = new_points
if verbose:
print('Rebuilding grid', end=' ')
grid = calculate(dbf,
comps,
active_phases,
output='GM',
model=models,
callables=callable_dict,
fake_points=True,
points=points_dict,
**grid_opts)
if verbose:
print('[{0} points, {1}]'.format(len(grid.points),
sizeof_fmt(grid.nbytes)),
end='\n')
properties.attrs['iterations'] += 1
# One last call to ensure 'properties' and 'grid' are consistent with one another
lower_convex_hull(grid, properties)
ravelled_X_view = grid['X'].values.view().reshape(
-1, grid['X'].values.shape[-1])
ravelled_Y_view = grid['Y'].values.view().reshape(
-1, grid['Y'].values.shape[-1])
ravelled_Phase_view = grid['Phase'].values.view().reshape(-1)
# Copy final point values from the grid and drop the index array
# For some reason direct construction doesn't work. We have to create empty and then assign.
properties['X'] = xray.DataArray(
np.empty_like(ravelled_X_view[properties['points'].values]),
dims=properties['points'].dims + ('component', ))
properties['X'].values[...] = ravelled_X_view[properties['points'].values]
properties['Y'] = xray.DataArray(
np.empty_like(ravelled_Y_view[properties['points'].values]),
dims=properties['points'].dims + ('internal_dof', ))
properties['Y'].values[...] = ravelled_Y_view[properties['points'].values]
# TODO: What about invariant reactions? We should perform a final driving force calculation here.
# We can handle that in the same post-processing step where we identify single-phase regions.
properties['Phase'] = xray.DataArray(np.empty_like(
ravelled_Phase_view[properties['points'].values]),
dims=properties['points'].dims)
properties['Phase'].values[...] = ravelled_Phase_view[
properties['points'].values]
del properties['points']
return properties
def calculate(dbf, comps, phases, mode=None, output='GM', fake_points=False, broadcast=True, parameters=None, **kwargs):
"""
Sample the property surface of 'output' containing the specified
components and phases. Model parameters are taken from 'dbf' and any
state variables (T, P, etc.) can be specified as keyword arguments.
Parameters
----------
dbf : Database
Thermodynamic database containing the relevant parameters.
comps : str or sequence
Names of components to consider in the calculation.
phases : str or sequence
Names of phases to consider in the calculation.
mode : string, optional
See 'make_callable' docstring for details.
output : string, optional
Model attribute to sample.
fake_points : bool, optional (Default: False)
If True, the first few points of the output surface will be fictitious
points used to define an equilibrium hyperplane guaranteed to be above
all the other points. This is used for convex hull computations.
broadcast : bool, optional
If True, broadcast given state variable lists against each other to create a grid.
If False, assume state variables are given as equal-length lists.
points : ndarray or a dict of phase names to ndarray, optional
Columns of ndarrays must be internal degrees of freedom (site fractions), sorted.
If this is not specified, points will be generated automatically.
pdens : int, a dict of phase names to int, or a seq of both, optional
Number of points to sample per degree of freedom.
Default: 2000; Default when called from equilibrium(): 500
model : Model, a dict of phase names to Model, or a seq of both, optional
Model class to use for each phase.
sampler : callable, a dict of phase names to callable, or a seq of both, optional
Function to sample phase constitution space.
Must have same signature as 'pycalphad.core.utils.point_sample'
grid_points : bool, a dict of phase names to bool, or a seq of both, optional (Default: True)
Whether to add evenly spaced points between end-members.
The density of points is determined by 'pdens'
parameters : dict, optional
Maps SymPy Symbol to numbers, for overriding the values of parameters in the Database.
Returns
-------
Dataset of the sampled attribute as a function of state variables
Examples
--------
None yet.
"""
# Here we check for any keyword arguments that are special, i.e.,
# there may be keyword arguments that aren't state variables
pdens_dict = unpack_kwarg(kwargs.pop('pdens', 2000), default_arg=2000)
points_dict = unpack_kwarg(kwargs.pop('points', None), default_arg=None)
model_dict = unpack_kwarg(kwargs.pop('model', Model), default_arg=Model)
callable_dict = unpack_kwarg(kwargs.pop('callables', None), default_arg=None)
mass_dict = unpack_kwarg(kwargs.pop('massfuncs', None), default_arg=None)
sampler_dict = unpack_kwarg(kwargs.pop('sampler', None), default_arg=None)
fixedgrid_dict = unpack_kwarg(kwargs.pop('grid_points', True), default_arg=True)
parameters = parameters or dict()
if isinstance(parameters, dict):
parameters = OrderedDict(sorted(parameters.items(), key=str))
param_symbols = tuple(parameters.keys())
param_values = np.atleast_1d(np.array(list(parameters.values()), dtype=np.float))
if isinstance(phases, str):
phases = [phases]
if isinstance(comps, (str, v.Species)):
comps = [comps]
comps = sorted(unpack_components(dbf, comps))
if points_dict is None and broadcast is False:
raise ValueError('The \'points\' keyword argument must be specified if broadcast=False is also given.')
nonvacant_components = [x for x in sorted(comps) if x.number_of_atoms > 0]
# Convert keyword strings to proper state variable objects
# If we don't do this, sympy will get confused during substitution
statevar_dict = dict((v.StateVariable(key), unpack_condition(value)) for (key, value) in kwargs.items())
# XXX: CompiledModel assumes P, T are the only state variables
if statevar_dict.get(v.P, None) is None:
statevar_dict[v.P] = 101325
if statevar_dict.get(v.T, None) is None:
statevar_dict[v.T] = 300
# Sort after default state variable check to fix gh-116
statevar_dict = collections.OrderedDict(sorted(statevar_dict.items(), key=lambda x: str(x[0])))
str_statevar_dict = collections.OrderedDict((str(key), unpack_condition(value)) \
for (key, value) in statevar_dict.items())
all_phase_data = []
comp_sets = {}
largest_energy = 1e30
maximum_internal_dof = 0
# Consider only the active phases
active_phases = dict((name.upper(), dbf.phases[name.upper()]) \
for name in unpack_phases(phases))
for phase_name, phase_obj in sorted(active_phases.items()):
# Build the symbolic representation of the energy
mod = model_dict[phase_name]
# if this is an object type, we need to construct it
if isinstance(mod, type):
try:
model_dict[phase_name] = mod = mod(dbf, comps, phase_name, parameters=parameters)
except DofError:
# we can't build the specified phase because the
# specified components aren't found in every sublattice
# we'll just skip it
warnings.warn("""Suspending specified phase {} due to
some sublattices containing only unspecified components""".format(phase_name))
continue
if points_dict[phase_name] is None:
maximum_internal_dof = max(maximum_internal_dof, sum(len(x) for x in mod.constituents))
else:
maximum_internal_dof = max(maximum_internal_dof, np.asarray(points_dict[phase_name]).shape[-1])
for phase_name, phase_obj in sorted(active_phases.items()):
try:
mod = model_dict[phase_name]
except KeyError:
continue
# this is a phase model we couldn't construct for whatever reason; skip it
if isinstance(mod, type):
continue
# Construct an ordered list of the variables
variables, sublattice_dof = generate_dof(phase_obj, mod.components)
# Build the "fast" representation of that model
if callable_dict[phase_name] is None:
try:
out = getattr(mod, output)
except AttributeError:
raise AttributeError('Missing Model attribute {0} specified for {1}'
.format(output, mod.__class__))
# As a last resort, treat undefined symbols as zero
# But warn the user when we do this
# This is consistent with TC's behavior
undefs = list(out.atoms(Symbol) - out.atoms(v.StateVariable))
for undef in undefs:
out = out.xreplace({undef: float(0)})
warnings.warn('Setting undefined symbol {0} for phase {1} to zero'.format(undef, phase_name))
comp_sets[phase_name] = build_functions(out, list(statevar_dict.keys()) + variables,
include_obj=True, include_grad=False,
parameters=param_symbols)
else:
comp_sets[phase_name] = callable_dict[phase_name]
if mass_dict[phase_name] is None:
pure_elements = [spec for spec in nonvacant_components
if (len(spec.constituents.keys()) == 1 and
list(spec.constituents.keys())[0] == spec.name)
]
# TODO: In principle, we should also check for undefs in mod.moles()
mass_dict[phase_name] = [build_functions(mod.moles(el), list(statevar_dict.keys()) + variables,
include_obj=True, include_grad=False,
parameters=param_symbols)
for el in pure_elements]
phase_record = PhaseRecord_from_cython(comps, list(statevar_dict.keys()) + variables,
np.array(dbf.phases[phase_name].sublattices, dtype=np.float),
param_values, comp_sets[phase_name], None, None, mass_dict[phase_name], None)
points = points_dict[phase_name]
if points is None:
points = _sample_phase_constitution(phase_name, phase_obj.constituents, sublattice_dof, comps,
tuple(variables), sampler_dict[phase_name] or point_sample,
fixedgrid_dict[phase_name], pdens_dict[phase_name])
points = np.atleast_2d(points)
fp = fake_points and (phase_name == sorted(active_phases.keys())[0])
phase_ds = _compute_phase_values(nonvacant_components, str_statevar_dict,
points, phase_record, output,
maximum_internal_dof, broadcast=broadcast,
largest_energy=float(largest_energy), fake_points=fp)
all_phase_data.append(phase_ds)
# speedup for single-phase case (found by profiling)
if len(all_phase_data) > 1:
final_ds = concat(all_phase_data, dim='points')
final_ds['points'].values = np.arange(len(final_ds['points']))
final_ds.coords['points'].values = np.arange(len(final_ds['points']))
else:
final_ds = all_phase_data[0]
return final_ds
def equilibrium(dbf,
comps,
phases,
conditions,
output=None,
model=None,
verbose=False,
broadcast=True,
calc_opts=None,
scheduler=dask.local.get_sync,
parameters=None,
**kwargs):
"""
Calculate the equilibrium state of a system containing the specified
components and phases, under the specified conditions.
Parameters
----------
dbf : Database
Thermodynamic database containing the relevant parameters.
comps : list
Names of components to consider in the calculation.
phases : list or dict
Names of phases to consider in the calculation.
conditions : dict or (list of dict)
StateVariables and their corresponding value.
output : str or list of str, optional
Additional equilibrium model properties (e.g., CPM, HM, etc.) to compute.
These must be defined as attributes in the Model class of each phase.
model : Model, a dict of phase names to Model, or a seq of both, optional
Model class to use for each phase.
verbose : bool, optional
Print details of calculations. Useful for debugging.
broadcast : bool
If True, broadcast conditions against each other. This will compute all combinations.
If False, each condition should be an equal-length list (or single-valued).
Disabling broadcasting is useful for calculating equilibrium at selected conditions,
when those conditions don't comprise a grid.
calc_opts : dict, optional
Keyword arguments to pass to `calculate`, the energy/property calculation routine.
scheduler : Dask scheduler, optional
Job scheduler for performing the computation.
If None, return a Dask graph of the computation instead of actually doing it.
parameters : dict, optional
Maps SymPy Symbol to numbers, for overriding the values of parameters in the Database.
Returns
-------
Structured equilibrium calculation, or Dask graph if scheduler=None.
Examples
--------
None yet.
"""
if not broadcast:
raise NotImplementedError('Broadcasting cannot yet be disabled')
from pycalphad import __version__ as pycalphad_version
phases = unpack_phases(phases) or sorted(dbf.phases.keys())
# remove phases that cannot be active
list_of_possible_phases = filter_phases(dbf, comps)
active_phases = sorted(
set(list_of_possible_phases).intersection(set(phases)))
if len(list_of_possible_phases) == 0:
raise ConditionError(
'There are no phases in the Database that can be active with components {0}'
.format(comps))
if len(active_phases) == 0:
raise ConditionError(
'None of the passed phases ({0}) are active. List of possible phases: {1}.'
.format(phases, list_of_possible_phases))
comps = sorted(comps)
if len(set(comps) - set(dbf.elements)) > 0:
raise EquilibriumError('Components not found in database: {}'.format(
','.join(set(comps) - set(dbf.elements))))
indep_vars = ['T', 'P']
calc_opts = calc_opts if calc_opts is not None else dict()
model = model if model is not None else FallbackModel
phase_records = dict()
diagnostic = kwargs.pop('_diagnostic', False)
callable_dict = kwargs.pop('callables', dict())
grad_callable_dict = kwargs.pop('grad_callables', dict())
hess_callable_dict = kwargs.pop('hess_callables', dict())
parameters = parameters if parameters is not None else dict()
if isinstance(parameters, dict):
parameters = OrderedDict(sorted(parameters.items(), key=str))
param_symbols = tuple(parameters.keys())
param_values = np.atleast_1d(
np.array(list(parameters.values()), dtype=np.float))
maximum_internal_dof = 0
# Modify conditions values to be within numerical limits, e.g., X(AL)=0
# Also wrap single-valued conditions with lists
conds = _adjust_conditions(conditions)
for cond in conds.keys():
if isinstance(cond,
(v.Composition,
v.ChemicalPotential)) and cond.species not in comps:
raise ConditionError(
'{} refers to non-existent component'.format(cond))
str_conds = OrderedDict((str(key), value) for key, value in conds.items())
num_calcs = np.prod([len(i) for i in str_conds.values()])
build_functions = compiled_build_functions
backend_mode = 'compiled'
if kwargs.get('_backend', None):
backend_mode = kwargs['_backend']
if verbose:
backend_dict = {
'compiled': 'Compiled (autowrap)',
'interpreted': 'Interpreted (autograd)'
}
print('Calculation Backend: {}'.format(
backend_dict.get(backend_mode, 'Custom')))
indep_vals = list([float(x) for x in np.atleast_1d(val)]
for key, val in str_conds.items() if key in indep_vars)
components = [x for x in sorted(comps) if not x.startswith('VA')]
# Construct models for each phase; prioritize user models
models = unpack_kwarg(model, default_arg=FallbackModel)
if verbose:
print('Components:', ' '.join(comps))
print('Phases:', end=' ')
max_phase_name_len = max(len(name) for name in active_phases)
# Need to allow for '_FAKE_' psuedo-phase
max_phase_name_len = max(max_phase_name_len, 6)
for name in active_phases:
mod = models[name]
if isinstance(mod, type):
models[name] = mod = mod(dbf, comps, name, parameters=parameters)
if isinstance(mod, CompiledModel):
phase_records[name.upper()] = PhaseRecord_from_compiledmodel(
mod, param_values)
maximum_internal_dof = max(maximum_internal_dof,
sum(mod.sublattice_dof))
else:
site_fracs = mod.site_fractions
variables = sorted(site_fracs, key=str)
maximum_internal_dof = max(maximum_internal_dof, len(site_fracs))
out = models[name].energy
if (not callable_dict.get(name, False)) or not (grad_callable_dict.get(name, False)) \
or (not hess_callable_dict.get(name, False)):
# Only force undefineds to zero if we're not overriding them
undefs = list(
out.atoms(Symbol) - out.atoms(v.StateVariable) -
set(param_symbols))
for undef in undefs:
out = out.xreplace({undef: float(0)})
cf, gf, hf = build_functions(out,
tuple([v.P, v.T] + site_fracs),
parameters=param_symbols)
if callable_dict.get(name, None) is None:
callable_dict[name] = cf
if grad_callable_dict.get(name, None) is None:
grad_callable_dict[name] = gf
if hess_callable_dict.get(name, None) is None:
hess_callable_dict[name] = hf
phase_records[name.upper()] = PhaseRecord_from_cython(
comps, variables,
np.array(dbf.phases[name].sublattices,
dtype=np.float), param_values, callable_dict[name],
grad_callable_dict[name], hess_callable_dict[name])
if verbose:
print(name, end=' ')
if verbose:
print('[done]', end='\n')
# 'calculate' accepts conditions through its keyword arguments
grid_opts = calc_opts.copy()
grid_opts.update(
{key: value
for key, value in str_conds.items() if key in indep_vars})
if 'pdens' not in grid_opts:
grid_opts['pdens'] = 500
coord_dict = str_conds.copy()
coord_dict['vertex'] = np.arange(len(components))
grid_shape = np.meshgrid(*coord_dict.values(), indexing='ij',
sparse=False)[0].shape
coord_dict['component'] = components
grid = delayed(calculate, pure=False)(dbf,
comps,
active_phases,
output='GM',
model=models,
callables=callable_dict,
fake_points=True,
parameters=parameters,
**grid_opts)
properties = delayed(Dataset, pure=False)(
{
'NP': (list(str_conds.keys()) + ['vertex'], np.empty(grid_shape)),
'GM': (list(str_conds.keys()), np.empty(grid_shape[:-1])),
'MU':
(list(str_conds.keys()) + ['component'], np.empty(grid_shape)),
'X': (list(str_conds.keys()) + ['vertex', 'component'],
np.empty(grid_shape + (grid_shape[-1], ))),
'Y': (list(str_conds.keys()) + ['vertex', 'internal_dof'],
np.empty(grid_shape + (maximum_internal_dof, ))),
'Phase': (list(str_conds.keys()) + ['vertex'],
np.empty(grid_shape, dtype='U%s' % max_phase_name_len)),
'points': (list(str_conds.keys()) + ['vertex'],
np.empty(grid_shape, dtype=np.int32))
},
coords=coord_dict,
attrs={
'engine': 'pycalphad %s' % pycalphad_version
},
)
# One last call to ensure 'properties' and 'grid' are consistent with one another
properties = delayed(lower_convex_hull, pure=False)(grid, properties)
conditions_per_chunk_per_axis = 2
if num_calcs > 1:
# Generate slices of 'properties'
slices = []
for val in grid_shape[:-1]:
idx_arr = list(range(val))
num_chunks = int(np.floor(val / conditions_per_chunk_per_axis))
if num_chunks > 0:
cond_slices = [
x for x in np.array_split(np.asarray(idx_arr), num_chunks)
if len(x) > 0
]
else:
cond_slices = [idx_arr]
slices.append(cond_slices)
chunk_dims = [len(slc) for slc in slices]
chunk_grid = np.array(
np.unravel_index(np.arange(np.prod(chunk_dims)), chunk_dims)).T
res = []
for chunk in chunk_grid:
prop_slice = properties[OrderedDict(
list(
zip(str_conds.keys(), [
np.atleast_1d(sl)[ch] for ch, sl in zip(chunk, slices)
])))]
job = delayed(_solve_eq_at_conditions,
pure=False)(comps, prop_slice, phase_records, grid,
list(str_conds.keys()), verbose)
res.append(job)
properties = delayed(_merge_property_slices,
pure=False)(properties, chunk_grid, slices,
list(str_conds.keys()), res)
else:
# Single-process job; don't create child processes
properties = delayed(_solve_eq_at_conditions,
pure=False)(comps, properties, phase_records,
grid, list(str_conds.keys()), verbose)
# Compute equilibrium values of any additional user-specified properties
output = output if isinstance(output, (list, tuple, set)) else [output]
# We already computed these properties so don't recompute them
output = sorted(set(output) - {'GM', 'MU'})
for out in output:
if (out is None) or (len(out) == 0):
continue
# TODO: How do we know if a specified property should be per_phase or not?
# For now, we make a best guess
if (out == 'degree_of_ordering') or (out == 'DOO'):
per_phase = True
else:
per_phase = False
for phase_name, mod in models.items():
if isinstance(mod, CompiledModel) or isinstance(
mod, FallbackModel):
models[phase_name] = Model(dbf,
comps,
phase_name,
parameters=parameters)
eqcal = delayed(_eqcalculate, pure=False)(dbf,
comps,
active_phases,
conditions,
out,
data=properties,
per_phase=per_phase,
model=models,
**calc_opts)
properties = delayed(properties.merge, pure=False)(eqcal,
inplace=True,
compat='equals')
if scheduler is not None:
properties = dask.compute(properties, get=scheduler)[0]
properties.attrs['created'] = datetime.utcnow().isoformat()
if len(kwargs) > 0:
warnings.warn(
'The following equilibrium keyword arguments were passed, but unused:\n{}'
.format(kwargs))
return properties
def _eqcalculate(dbf,
comps,
phases,
conditions,
output,
data=None,
per_phase=False,
**kwargs):
"""
WARNING: API/calling convention not finalized.
Compute the *equilibrium value* of a property.
This function differs from `calculate` in that it computes
thermodynamic equilibrium instead of randomly sampling the
internal degrees of freedom of a phase.
Because of that, it's slower than `calculate`.
This plugs in the equilibrium phase and site fractions
to compute a thermodynamic property defined in a Model.
Parameters
----------
dbf : Database
Thermodynamic database containing the relevant parameters.
comps : list
Names of components to consider in the calculation.
phases : list or dict
Names of phases to consider in the calculation.
conditions : dict or (list of dict)
StateVariables and their corresponding value.
output : str
Equilibrium model property (e.g., CPM, HM, etc.) to compute.
This must be defined as an attribute in the Model class of each phase.
data : Dataset, optional
Previous result of call to `equilibrium`.
Should contain the equilibrium configurations at the conditions of interest.
If the databases are not the same as in the original calculation,
the results may be meaningless. If None, `equilibrium` will be called.
Specifying this keyword argument can save the user some time if several properties
need to be calculated in succession.
per_phase : bool, optional
If True, compute and return the property for each phase present.
If False, return the total system value, weighted by the phase fractions.
kwargs
Passed to `calculate`.
Returns
-------
Dataset of property as a function of equilibrium conditions
"""
if data is None:
data = equilibrium(dbf, comps, phases, conditions)
active_phases = unpack_phases(phases) or sorted(dbf.phases.keys())
conds = _adjust_conditions(conditions)
indep_vars = ['P', 'T']
# TODO: Rewrite this to use the coord dict from 'data'
str_conds = OrderedDict((str(key), value) for key, value in conds.items())
indep_vals = list([float(x) for x in np.atleast_1d(val)]
for key, val in str_conds.items() if key in indep_vars)
coord_dict = str_conds.copy()
components = [x for x in sorted(comps) if not x.startswith('VA')]
coord_dict['vertex'] = np.arange(len(components))
grid_shape = np.meshgrid(*coord_dict.values(), indexing='ij',
sparse=False)[0].shape
prop_shape = grid_shape
prop_dims = list(str_conds.keys()) + ['vertex']
result = Dataset({output: (prop_dims, np.full(prop_shape, np.nan))},
coords=coord_dict)
# For each phase select all conditions where that phase exists
# Perform the appropriate calculation and then write the result back
for phase in active_phases:
dof = sum([len(x) for x in dbf.phases[phase].constituents])
current_phase_indices = (data.Phase.values == phase)
if ~np.any(current_phase_indices):
continue
points = data.Y.values[np.nonzero(current_phase_indices)][..., :dof]
statevar_indices = np.nonzero(current_phase_indices)[:len(indep_vals)]
statevars = {
key: np.take(np.asarray(vals), idx)
for key, vals, idx in zip(indep_vars, indep_vals, statevar_indices)
}
statevars.update(kwargs)
if statevars.get('mode', None) is None:
statevars['mode'] = 'numpy'
calcres = calculate(dbf,
comps, [phase],
output=output,
points=points,
broadcast=False,
**statevars)
result[output].values[np.nonzero(
current_phase_indices)] = calcres[output].values
if not per_phase:
result[output] = (result[output] * data['NP']).sum(dim='vertex',
skipna=True)
else:
result['Phase'] = data['Phase'].copy()
result['NP'] = data['NP'].copy()
return result
def _eqcalculate(dbf, comps, phases, conditions, output, data=None, per_phase=False, callables=None, parameters=None,
**kwargs):
"""
WARNING: API/calling convention not finalized.
Compute the *equilibrium value* of a property.
This function differs from `calculate` in that it computes
thermodynamic equilibrium instead of randomly sampling the
internal degrees of freedom of a phase.
Because of that, it's slower than `calculate`.
This plugs in the equilibrium phase and site fractions
to compute a thermodynamic property defined in a Model.
Parameters
----------
dbf : Database
Thermodynamic database containing the relevant parameters.
comps : list
Names of components to consider in the calculation.
phases : list or dict
Names of phases to consider in the calculation.
conditions : dict or (list of dict)
StateVariables and their corresponding value.
output : str
Equilibrium model property (e.g., CPM, HM, etc.) to compute.
This must be defined as an attribute in the Model class of each phase.
data : Dataset, optional
Previous result of call to `equilibrium`.
Should contain the equilibrium configurations at the conditions of interest.
If the databases are not the same as in the original calculation,
the results may be meaningless. If None, `equilibrium` will be called.
Specifying this keyword argument can save the user some time if several properties
need to be calculated in succession.
per_phase : bool, optional
If True, compute and return the property for each phase present.
If False, return the total system value, weighted by the phase fractions.
parameters : dict, optional
Maps SymPy Symbol to numbers, for overriding the values of parameters in the Database.
callables : dict
Callable functions to compute 'output' for each phase.
kwargs
Passed to `calculate`.
Returns
-------
Dataset of property as a function of equilibrium conditions
"""
if data is None:
data = equilibrium(dbf, comps, phases, conditions)
active_phases = unpack_phases(phases) or sorted(dbf.phases.keys())
conds = _adjust_conditions(conditions)
indep_vars = ['N', 'P', 'T']
# TODO: Rewrite this to use the coord dict from 'data'
str_conds = OrderedDict((str(key), value) for key, value in conds.items())
indep_vals = list([float(x) for x in np.atleast_1d(val)]
for key, val in str_conds.items() if key in indep_vars)
coord_dict = str_conds.copy()
components = [x for x in sorted(comps)]
desired_active_pure_elements = [list(x.constituents.keys()) for x in components]
desired_active_pure_elements = [el.upper() for constituents in desired_active_pure_elements for el in constituents]
pure_elements = sorted(set([x for x in desired_active_pure_elements if x != 'VA']))
coord_dict['vertex'] = np.arange(len(pure_elements) + 1) # +1 is to accommodate the degenerate degree of freedom at the invariant reactions
grid_shape = np.meshgrid(*coord_dict.values(),
indexing='ij', sparse=False)[0].shape
prop_shape = grid_shape
prop_dims = list(str_conds.keys()) + ['vertex']
result = Dataset({output: (prop_dims, np.full(prop_shape, np.nan))}, coords=coord_dict)
# For each phase select all conditions where that phase exists
# Perform the appropriate calculation and then write the result back
for phase in active_phases:
dof = sum([len(x) for x in dbf.phases[phase].constituents])
current_phase_indices = (data.Phase.values == phase)
if ~np.any(current_phase_indices):
continue
points = data.Y.values[np.nonzero(current_phase_indices)][..., :dof]
statevar_indices = np.nonzero(current_phase_indices)[:len(indep_vals)]
statevars = {key: np.take(np.asarray(vals), idx)
for key, vals, idx in zip(indep_vars, indep_vals, statevar_indices)}
statevars.update(kwargs)
if statevars.get('mode', None) is None:
statevars['mode'] = 'numpy'
calcres = calculate(dbf, comps, [phase], output=output, points=points, broadcast=False,
callables=callables, parameters=parameters, **statevars)
result[output].values[np.nonzero(current_phase_indices)] = calcres[output].values
if not per_phase:
result[output] = (result[output] * data['NP']).sum(dim='vertex', skipna=True)
else:
result['Phase'] = data['Phase'].copy()
result['NP'] = data['NP'].copy()
return result
If None, return a Dask graph of the computation instead of actually doing it.
parameters : dict, optional
Maps SymPy Symbol to numbers, for overriding the values of parameters in the Database.
Returns
-------
Structured equilibrium calculation, or Dask graph if scheduler=None.
Examples
--------
None yet.
"""
if not broadcast:
raise NotImplementedError('Broadcasting cannot yet be disabled')
from pycalphad import __version__ as pycalphad_version
active_phases = unpack_phases(phases) or sorted(dbf.phases.keys())
comps = sorted(comps)
if len(set(comps) - set(dbf.elements)) > 0:
raise EquilibriumError('Components not found in database: {}'.format(
','.join(set(comps) - set(dbf.elements))))
indep_vars = ['T', 'P']
calc_opts = calc_opts if calc_opts is not None else dict()
model = model if model is not None else FallbackModel
phase_records = dict()
diagnostic = kwargs.pop('_diagnostic', False)
callable_dict = kwargs.pop('callables', dict())
grad_callable_dict = kwargs.pop('grad_callables', dict())
hess_callable_dict = kwargs.pop('hess_callables', dict())
parameters = parameters if parameters is not None else dict()
if isinstance(parameters, dict):
parameters = OrderedDict(sorted(parameters.items(), key=str))
def equilibrium(dbf, comps, phases, conditions, **kwargs):
"""
Calculate the equilibrium state of a system containing the specified
components and phases, under the specified conditions.
Model parameters are taken from 'dbf'.
Parameters
----------
dbf : Database
Thermodynamic database containing the relevant parameters.
comps : list
Names of components to consider in the calculation.
phases : list or dict
Names of phases to consider in the calculation.
conditions : dict or (list of dict)
StateVariables and their corresponding value.
verbose : bool, optional (Default: True)
Show progress of calculations.
grid_opts : dict, optional
Keyword arguments to pass to the initial grid routine.
Returns
-------
Structured equilibrium calculation.
Examples
--------
None yet.
"""
active_phases = unpack_phases(phases) or sorted(dbf.phases.keys())
comps = sorted(comps)
indep_vars = ['T', 'P']
grid_opts = kwargs.pop('grid_opts', dict())
verbose = kwargs.pop('verbose', True)
phase_records = dict()
callable_dict = kwargs.pop('callables', dict())
grad_callable_dict = kwargs.pop('grad_callables', dict())
points_dict = dict()
maximum_internal_dof = 0
# Construct models for each phase; prioritize user models
models = unpack_kwarg(kwargs.pop('model', Model), default_arg=Model)
if verbose:
print('Components:', ' '.join(comps))
print('Phases:', end=' ')
for name in active_phases:
mod = models[name]
if isinstance(mod, type):
models[name] = mod = mod(dbf, comps, name)
variables = sorted(mod.energy.atoms(v.StateVariable).union(
{key
for key in conditions.keys() if key in [v.T, v.P]}),
key=str)
site_fracs = sorted(mod.energy.atoms(v.SiteFraction), key=str)
maximum_internal_dof = max(maximum_internal_dof, len(site_fracs))
# Extra factor '1e-100...' is to work around an annoying broadcasting bug for zero gradient entries
models[name].models['_broadcaster'] = 1e-100 * Mul(*variables)**3
out = models[name].energy
if name not in callable_dict:
undefs = list(out.atoms(Symbol) - out.atoms(v.StateVariable))
for undef in undefs:
out = out.xreplace({undef: float(0)})
# callable_dict takes variables in a different order due to calculate() pecularities
callable_dict[name] = make_callable(
out,
sorted((key for key in conditions.keys() if key in [v.T, v.P]),
key=str) + site_fracs)
if name not in grad_callable_dict:
grad_func = make_callable(
Matrix([out]).jacobian(variables), variables)
else:
grad_func = grad_callable_dict[name]
# Adjust gradient by the approximate chemical potentials
plane_vars = sorted(models[name].energy.atoms(v.SiteFraction), key=str)
hyperplane = Add(*[
v.MU(i) * mole_fraction(dbf.phases[name], comps, i) for i in comps
if i != 'VA'
])
# Workaround an annoying bug with zero gradient entries
# This forces numerically zero entries to broadcast correctly
hyperplane += 1e-100 * Mul(*([v.MU(i) for i in comps if i != 'VA'] +
plane_vars + [v.T, v.P]))**3
plane_grad = make_callable(
Matrix([hyperplane]).jacobian(variables),
[v.MU(i) for i in comps if i != 'VA'] + plane_vars + [v.T, v.P])
plane_hess = make_callable(hessian(hyperplane, variables),
[v.MU(i) for i in comps if i != 'VA'] +
plane_vars + [v.T, v.P])
phase_records[name.upper()] = PhaseRecord(variables=variables,
grad=grad_func,
plane_grad=plane_grad,
plane_hess=plane_hess)
if verbose:
print(name, end=' ')
if verbose:
print('[done]', end='\n')
conds = OrderedDict((key, unpack_condition(value))
for key, value in sorted(conditions.items(), key=str))
str_conds = OrderedDict((str(key), value) for key, value in conds.items())
indep_vals = list([float(x) for x in np.atleast_1d(val)]
for key, val in str_conds.items() if key in indep_vars)
components = [x for x in sorted(comps) if not x.startswith('VA')]
# 'calculate' accepts conditions through its keyword arguments
grid_opts.update(
{key: value
for key, value in str_conds.items() if key in indep_vars})
if 'pdens' not in grid_opts:
grid_opts['pdens'] = 10
coord_dict = str_conds.copy()
coord_dict['vertex'] = np.arange(len(components))
grid_shape = np.meshgrid(*coord_dict.values(), indexing='ij',
sparse=False)[0].shape
coord_dict['component'] = components
if verbose:
print('Computing initial grid', end=' ')
grid = calculate(dbf,
comps,
active_phases,
output='GM',
model=models,
callables=callable_dict,
fake_points=True,
**grid_opts)
if verbose:
print('[{0} points, {1}]'.format(len(grid.points),
sizeof_fmt(grid.nbytes)),
end='\n')
properties = xray.Dataset(
{
'NP': (list(str_conds.keys()) + ['vertex'], np.empty(grid_shape)),
'GM': (list(str_conds.keys()), np.empty(grid_shape[:-1])),
'MU':
(list(str_conds.keys()) + ['component'], np.empty(grid_shape)),
'points': (list(str_conds.keys()) + ['vertex'],
np.empty(grid_shape, dtype=np.int))
},
coords=coord_dict,
attrs={'iterations': 1},
)
# Store the potentials from the previous iteration
current_potentials = properties.MU.copy()
for iteration in range(MAX_ITERATIONS):
if verbose:
print('Computing convex hull [iteration {}]'.format(
properties.attrs['iterations']))
# lower_convex_hull will modify properties
lower_convex_hull(grid, properties)
progress = np.abs(current_potentials - properties.MU).max().values
if verbose:
print('progress', progress)
if progress < MIN_PROGRESS:
if verbose:
print('Convergence achieved')
break
current_potentials[...] = properties.MU.values
if verbose:
print('Refining convex hull')
# Insert extra dimensions for non-T,P conditions so GM broadcasts correctly
energy_broadcast_shape = grid.GM.values.shape[:len(indep_vals)] + \
(1,) * (len(str_conds) - len(indep_vals)) + (grid.GM.values.shape[-1],)
driving_forces = np.einsum('...i,...i',
properties.MU.values[..., np.newaxis, :],
grid.X.values[np.index_exp[...] +
(np.newaxis,) * (len(str_conds) - len(indep_vals)) +
np.index_exp[:, :]]) - \
grid.GM.values.view().reshape(energy_broadcast_shape)
for name in active_phases:
dof = len(models[name].energy.atoms(v.SiteFraction))
current_phase_indices = (grid.Phase.values == name
).reshape(energy_broadcast_shape[:-1] +
(-1, ))
# Broadcast to capture all conditions
current_phase_indices = np.broadcast_arrays(
current_phase_indices, np.empty(driving_forces.shape))[0]
# This reshape is safe as long as phases have the same number of points at all indep. conditions
current_phase_driving_forces = driving_forces[
current_phase_indices].reshape(
current_phase_indices.shape[:-1] + (-1, ))
# Note: This works as long as all points are in the same phase order for all T, P
current_site_fractions = grid.Y.values[..., current_phase_indices[
(0, ) * len(str_conds)], :]
if np.sum(
current_site_fractions[(0, ) *
len(indep_vals)][..., :dof]) == dof:
# All site fractions are 1, aka zero internal degrees of freedom
# Impossible to refine these points, so skip this phase
points_dict[name] = current_site_fractions[
(0, ) * len(indep_vals)][..., :dof]
continue
# Find the N points with largest driving force for a given set of conditions
# Remember that driving force has a sign, so we want the "most positive" values
# N is the number of components, in this context
# N points define a 'best simplex' for every set of conditions
# We also need to restrict ourselves to one phase at a time
trial_indices = np.argpartition(current_phase_driving_forces,
-len(components),
axis=-1)[..., -len(components):]
trial_indices = trial_indices.ravel()
statevar_indices = np.unravel_index(
np.arange(
np.multiply.reduce(properties.GM.values.shape +
(len(components), ))),
properties.GM.values.shape +
(len(components), ))[:len(indep_vals)]
points = current_site_fractions[np.index_exp[statevar_indices +
(trial_indices, )]]
points.shape = properties.points.shape[:-1] + (
-1, maximum_internal_dof)
# The Y arrays have been padded, so we should slice off the padding
points = points[..., :dof]
# Workaround for derivative issues at endmembers
points[points == 0.] = MIN_SITE_FRACTION
if len(points) == 0:
if name in points_dict:
del points_dict[name]
# No nearly stable points: skip this phase
continue
num_vars = len(phase_records[name].variables)
plane_grad = phase_records[name].plane_grad
plane_hess = phase_records[name].plane_hess
statevar_grid = np.meshgrid(*itertools.chain(indep_vals),
sparse=True,
indexing='xy')
# TODO: A more sophisticated treatment of constraints
num_constraints = len(indep_vals) + len(
dbf.phases[name].sublattices)
constraint_jac = np.zeros((num_constraints, num_vars))
# Independent variables are always fixed (in this limited implementation)
for idx in range(len(indep_vals)):
constraint_jac[idx, idx] = 1
# This is for site fraction balance constraints
var_idx = len(indep_vals)
for idx in range(len(dbf.phases[name].sublattices)):
active_in_subl = set(
dbf.phases[name].constituents[idx]).intersection(comps)
constraint_jac[len(indep_vals) + idx,
var_idx:var_idx + len(active_in_subl)] = 1
var_idx += len(active_in_subl)
grad = phase_records[name].grad(
*itertools.chain(statevar_grid, points.T))
if grad.dtype == 'object':
# Workaround a bug in zero gradient entries
grad_zeros = np.zeros(points.T.shape[1:], dtype=np.float)
for i in np.arange(grad.shape[0]):
if isinstance(grad[i], int):
grad[i] = grad_zeros
grad = np.array(grad.tolist(), dtype=np.float)
bcasts = np.broadcast_arrays(
*itertools.chain(properties.MU.values.T, points.T))
cast_grad = -plane_grad(*itertools.chain(bcasts, [0], [0]))
cast_grad = cast_grad.T + grad.T
grad = cast_grad
grad.shape = grad.shape[:-1] # Remove extraneous dimension
# This Hessian is an approximation updated using the BFGS method
# See Nocedal and Wright, ch.3, p. 198
# Initialize as identity matrix
hess = broadcast_to(np.eye(num_vars),
grad.shape + (grad.shape[-1], )).copy()
newton_iteration = 0
while newton_iteration < MAX_NEWTON_ITERATIONS:
e_matrix = np.linalg.inv(hess)
dy_unconstrained = -np.einsum('...ij,...j->...i', e_matrix,
grad)
proj_matrix = np.dot(e_matrix, constraint_jac.T)
inv_matrix = np.rollaxis(np.dot(constraint_jac, proj_matrix),
0, -1)
inv_term = np.linalg.inv(inv_matrix)
first_term = np.einsum('...ij,...jk->...ik', proj_matrix,
inv_term)
# Normally a term for the residual here
# We only choose starting points which obey the constraints, so r = 0
cons_summation = np.einsum('...i,...ji->...j',
dy_unconstrained, constraint_jac)
cons_correction = np.einsum('...ij,...j->...i', first_term,
cons_summation)
dy_constrained = dy_unconstrained - cons_correction
# TODO: Support for adaptive changing independent variable steps
new_direction = dy_constrained[..., len(indep_vals):]
# Backtracking line search
new_points = points + INITIAL_STEP_SIZE * new_direction
alpha = np.full(new_points.shape[:-1],
INITIAL_STEP_SIZE,
dtype=np.float)
negative_points = np.any(new_points < 0., axis=-1)
while np.any(negative_points):
alpha[negative_points] *= 0.1
new_points = points + alpha[...,
np.newaxis] * new_direction
negative_points = np.any(new_points < 0., axis=-1)
# If we made "near" zero progress on any points, don't update the Hessian until
# we've rebuilt the convex hull
# Nocedal and Wright recommend against skipping Hessian updates
# They recommend using a damped update approach, pp. 538-539 of their book
# TODO: Check the projected gradient norm, not the step length
if np.any(
np.max(np.abs(alpha[..., np.newaxis] * new_direction),
axis=-1) < MIN_STEP_LENGTH):
break
# Workaround for derivative issues at endmembers
new_points[new_points == 0.] = 1e-16
# BFGS update to Hessian
new_grad = phase_records[name].grad(
*itertools.chain(statevar_grid, new_points.T))
if new_grad.dtype == 'object':
# Workaround a bug in zero gradient entries
grad_zeros = np.zeros(new_points.T.shape[1:],
dtype=np.float)
for i in np.arange(new_grad.shape[0]):
if isinstance(new_grad[i], int):
new_grad[i] = grad_zeros
new_grad = np.array(new_grad.tolist(), dtype=np.float)
bcasts = np.broadcast_arrays(
*itertools.chain(properties.MU.values.T, new_points.T))
cast_grad = -plane_grad(*itertools.chain(bcasts, [0], [0]))
cast_grad = cast_grad.T + new_grad.T
new_grad = cast_grad
new_grad.shape = new_grad.shape[:
-1] # Remove extraneous dimension
# Notation used here consistent with Nocedal and Wright
s_k = np.empty(points.shape[:-1] +
(points.shape[-1] + len(indep_vals), ))
# Zero out independent variable changes for now
s_k[..., :len(indep_vals)] = 0
s_k[..., len(indep_vals):] = new_points - points
y_k = new_grad - grad
s_s_term = np.einsum('...j,...k->...jk', s_k, s_k)
s_b_s_term = np.einsum('...i,...ij,...j', s_k, hess, s_k)
y_y_y_s_term = np.einsum('...j,...k->...jk', y_k, y_k) / \
np.einsum('...i,...i', y_k, s_k)[..., np.newaxis, np.newaxis]
update = np.einsum('...ij,...jk,...kl->...il', hess, s_s_term, hess) / \
s_b_s_term[..., np.newaxis, np.newaxis] + y_y_y_s_term
hess = hess - update
cast_hess = -plane_hess(
*itertools.chain(bcasts, [0], [0])).T + hess
hess = -cast_hess #TODO: Why does this fix things?
# TODO: Verify that the chosen step lengths reduce the energy
points = new_points
grad = new_grad
newton_iteration += 1
new_points = new_points.reshape(
new_points.shape[:len(indep_vals)] +
(-1, new_points.shape[-1]))
new_points = np.concatenate(
(current_site_fractions[..., :dof], new_points), axis=-2)
points_dict[name] = new_points
if verbose:
print('Rebuilding grid', end=' ')
grid = calculate(dbf,
comps,
active_phases,
output='GM',
model=models,
callables=callable_dict,
fake_points=True,
points=points_dict,
**grid_opts)
if verbose:
print('[{0} points, {1}]'.format(len(grid.points),
sizeof_fmt(grid.nbytes)),
end='\n')
properties.attrs['iterations'] += 1
# One last call to ensure 'properties' and 'grid' are consistent with one another
lower_convex_hull(grid, properties)
ravelled_X_view = grid['X'].values.view().reshape(
-1, grid['X'].values.shape[-1])
ravelled_Y_view = grid['Y'].values.view().reshape(
-1, grid['Y'].values.shape[-1])
ravelled_Phase_view = grid['Phase'].values.view().reshape(-1)
# Copy final point values from the grid and drop the index array
# For some reason direct construction doesn't work. We have to create empty and then assign.
properties['X'] = xray.DataArray(
np.empty_like(ravelled_X_view[properties['points'].values]),
dims=properties['points'].dims + ('component', ))
properties['X'].values[...] = ravelled_X_view[properties['points'].values]
properties['Y'] = xray.DataArray(
np.empty_like(ravelled_Y_view[properties['points'].values]),
dims=properties['points'].dims + ('internal_dof', ))
properties['Y'].values[...] = ravelled_Y_view[properties['points'].values]
# TODO: What about invariant reactions? We should perform a final driving force calculation here.
# We can handle that in the same post-processing step where we identify single-phase regions.
properties['Phase'] = xray.DataArray(np.empty_like(
ravelled_Phase_view[properties['points'].values]),
dims=properties['points'].dims)
properties['Phase'].values[...] = ravelled_Phase_view[
properties['points'].values]
del properties['points']
return properties
def equilibrium(dbf, comps, phases, conditions, output=None, model=None,
verbose=False, broadcast=True, calc_opts=None,
scheduler='sync', parameters=None, solver=None, callables=None,
**kwargs):
"""
Calculate the equilibrium state of a system containing the specified
components and phases, under the specified conditions.
Parameters
----------
dbf : Database
Thermodynamic database containing the relevant parameters.
comps : list
Names of components to consider in the calculation.
phases : list or dict
Names of phases to consider in the calculation.
conditions : dict or (list of dict)
StateVariables and their corresponding value.
output : str or list of str, optional
Additional equilibrium model properties (e.g., CPM, HM, etc.) to compute.
These must be defined as attributes in the Model class of each phase.
model : Model, a dict of phase names to Model, or a seq of both, optional
Model class to use for each phase.
verbose : bool, optional
Print details of calculations. Useful for debugging.
broadcast : bool
If True, broadcast conditions against each other. This will compute all combinations.
If False, each condition should be an equal-length list (or single-valued).
Disabling broadcasting is useful for calculating equilibrium at selected conditions,
when those conditions don't comprise a grid.
calc_opts : dict, optional
Keyword arguments to pass to `calculate`, the energy/property calculation routine.
scheduler : Dask scheduler, optional
Job scheduler for performing the computation.
If None, return a Dask graph of the computation instead of actually doing it.
parameters : dict, optional
Maps SymPy Symbol to numbers, for overriding the values of parameters in the Database.
solver : pycalphad.core.solver.SolverBase
Instance of a solver that is used to calculate local equilibria.
Defaults to a pycalphad.core.solver.InteriorPointSolver.
callables : dict, optional
Pre-computed callable functions for equilibrium calculation.
Returns
-------
Structured equilibrium calculation, or Dask graph if scheduler=None.
Examples
--------
None yet.
"""
if not broadcast:
raise NotImplementedError('Broadcasting cannot yet be disabled')
comps = sorted(unpack_components(dbf, comps))
phases = unpack_phases(phases) or sorted(dbf.phases.keys())
# remove phases that cannot be active
list_of_possible_phases = filter_phases(dbf, comps)
active_phases = sorted(set(list_of_possible_phases).intersection(set(phases)))
if len(list_of_possible_phases) == 0:
raise ConditionError('There are no phases in the Database that can be active with components {0}'.format(comps))
if len(active_phases) == 0:
raise ConditionError('None of the passed phases ({0}) are active. List of possible phases: {1}.'.format(phases, list_of_possible_phases))
if isinstance(comps, (str, v.Species)):
comps = [comps]
if len(set(comps) - set(dbf.species)) > 0:
raise EquilibriumError('Components not found in database: {}'
.format(','.join([c.name for c in (set(comps) - set(dbf.species))])))
calc_opts = calc_opts if calc_opts is not None else dict()
solver = solver if solver is not None else InteriorPointSolver(verbose=verbose)
parameters = parameters if parameters is not None else dict()
if isinstance(parameters, dict):
parameters = OrderedDict(sorted(parameters.items(), key=str))
models = instantiate_models(dbf, comps, active_phases, model=model, parameters=parameters)
# Temporary solution until constraint system improves
if conditions.get(v.N) is None:
conditions[v.N] = 1
if np.any(np.array(conditions[v.N]) != 1):
raise ConditionError('N!=1 is not yet supported, got N={}'.format(conditions[v.N]))
# Modify conditions values to be within numerical limits, e.g., X(AL)=0
# Also wrap single-valued conditions with lists
conds = _adjust_conditions(conditions)
for cond in conds.keys():
if isinstance(cond, (v.Composition, v.ChemicalPotential)) and cond.species not in comps:
raise ConditionError('{} refers to non-existent component'.format(cond))
state_variables = sorted(get_state_variables(models=models, conds=conds), key=str)
str_conds = OrderedDict((str(key), value) for key, value in conds.items())
num_calcs = np.prod([len(i) for i in str_conds.values()])
components = [x for x in sorted(comps)]
desired_active_pure_elements = [list(x.constituents.keys()) for x in components]
desired_active_pure_elements = [el.upper() for constituents in desired_active_pure_elements for el in constituents]
pure_elements = sorted(set([x for x in desired_active_pure_elements if x != 'VA']))
if verbose:
print('Components:', ' '.join([str(x) for x in comps]))
print('Phases:', end=' ')
output = output if output is not None else 'GM'
output = output if isinstance(output, (list, tuple, set)) else [output]
output = set(output)
output |= {'GM'}
output = sorted(output)
need_hessians = any(type(c) in v.CONDITIONS_REQUIRING_HESSIANS for c in conds.keys())
phase_records = build_phase_records(dbf, comps, active_phases, conds, models,
output='GM', callables=callables,
parameters=parameters, verbose=verbose,
build_gradients=True, build_hessians=need_hessians)
if verbose:
print('[done]', end='\n')
# 'calculate' accepts conditions through its keyword arguments
grid_opts = calc_opts.copy()
statevar_strings = [str(x) for x in state_variables]
grid_opts.update({key: value for key, value in str_conds.items() if key in statevar_strings})
if 'pdens' not in grid_opts:
grid_opts['pdens'] = 500
grid = delayed(calculate, pure=False)(dbf, comps, active_phases,
model=models, fake_points=True,
callables=callables, output='GM',
parameters=parameters, **grid_opts)
coord_dict = str_conds.copy()
coord_dict['vertex'] = np.arange(
len(pure_elements) + 1) # +1 is to accommodate the degenerate degree of freedom at the invariant reactions
coord_dict['component'] = pure_elements
grid_shape = tuple(len(x) for x in conds.values()) + (len(pure_elements)+1,)
properties = delayed(starting_point, pure=False)(conds, state_variables, phase_records, grid)
conditions_per_chunk_per_axis = 2
if num_calcs > 1:
# Generate slices of 'properties'
slices = []
for val in grid_shape[:-1]:
idx_arr = list(range(val))
num_chunks = int(np.floor(val/conditions_per_chunk_per_axis))
if num_chunks > 0:
cond_slices = [x for x in np.array_split(np.asarray(idx_arr), num_chunks) if len(x) > 0]
else:
cond_slices = [idx_arr]
slices.append(cond_slices)
chunk_dims = [len(slc) for slc in slices]
chunk_grid = np.array(np.unravel_index(np.arange(np.prod(chunk_dims)), chunk_dims)).T
res = []
for chunk in chunk_grid:
prop_slice = properties[OrderedDict(list(zip(str_conds.keys(),
[np.atleast_1d(sl)[ch] for ch, sl in zip(chunk, slices)])))]
job = delayed(_solve_eq_at_conditions, pure=False)(comps, prop_slice, phase_records, grid,
list(str_conds.keys()), state_variables, verbose, solver=solver)
res.append(job)
properties = delayed(_merge_property_slices, pure=False)(properties, chunk_grid, slices, list(str_conds.keys()), res)
else:
# Single-process job; don't create child processes
properties = delayed(_solve_eq_at_conditions, pure=False)(comps, properties, phase_records, grid,
list(str_conds.keys()), state_variables, verbose, solver=solver)
# Compute equilibrium values of any additional user-specified properties
# We already computed these properties so don't recompute them
output = sorted(set(output) - {'GM', 'MU'})
for out in output:
if (out is None) or (len(out) == 0):
continue
# TODO: How do we know if a specified property should be per_phase or not?
# For now, we make a best guess
if (out == 'degree_of_ordering') or (out == 'DOO'):
per_phase = True
else:
per_phase = False
eqcal = delayed(_eqcalculate, pure=False)(dbf, comps, active_phases, conditions, out,
data=properties, per_phase=per_phase,
callables=callables,
parameters=parameters,
model=models, **calc_opts)
properties = delayed(properties.merge, pure=False)(eqcal, compat='equals')
if scheduler is not None:
properties = dask.compute(properties, scheduler=scheduler)[0]
properties.attrs['created'] = datetime.utcnow().isoformat()
if len(kwargs) > 0:
warnings.warn('The following equilibrium keyword arguments were passed, but unused:\n{}'.format(kwargs))
return properties
def calculate(dbf, comps, phases, mode=None, output='GM', fake_points=False, broadcast=True, tmpman=None, **kwargs):
"""
Sample the property surface of 'output' containing the specified
components and phases. Model parameters are taken from 'dbf' and any
state variables (T, P, etc.) can be specified as keyword arguments.
Parameters
----------
dbf : Database
Thermodynamic database containing the relevant parameters.
comps : str or sequence
Names of components to consider in the calculation.
phases : str or sequence
Names of phases to consider in the calculation.
mode : string, optional
See 'make_callable' docstring for details.
output : string, optional
Model attribute to sample.
fake_points : bool, optional (Default: False)
If True, the first few points of the output surface will be fictitious
points used to define an equilibrium hyperplane guaranteed to be above
all the other points. This is used for convex hull computations.
broadcast : bool, optional
If True, broadcast given state variable lists against each other to create a grid.
If False, assume state variables are given as equal-length lists.
tmpman : TempfileManager, optional
Context manager for temporary file creation during the calculation.
points : ndarray or a dict of phase names to ndarray, optional
Columns of ndarrays must be internal degrees of freedom (site fractions), sorted.
If this is not specified, points will be generated automatically.
pdens : int, a dict of phase names to int, or a seq of both, optional
Number of points to sample per degree of freedom.
model : Model, a dict of phase names to Model, or a seq of both, optional
Model class to use for each phase.
sampler : callable, a dict of phase names to callable, or a seq of both, optional
Function to sample phase constitution space.
Must have same signature as 'pycalphad.core.utils.point_sample'
grid_points : bool, a dict of phase names to bool, or a seq of both, optional (Default: True)
Whether to add evenly spaced points between end-members.
The density of points is determined by 'pdens'
Returns
-------
Dataset of the sampled attribute as a function of state variables
Examples
--------
None yet.
"""
# Here we check for any keyword arguments that are special, i.e.,
# there may be keyword arguments that aren't state variables
pdens_dict = unpack_kwarg(kwargs.pop('pdens', 2000), default_arg=2000)
points_dict = unpack_kwarg(kwargs.pop('points', None), default_arg=None)
model_dict = unpack_kwarg(kwargs.pop('model', Model), default_arg=Model)
callable_dict = unpack_kwarg(kwargs.pop('callables', None), default_arg=None)
sampler_dict = unpack_kwarg(kwargs.pop('sampler', None), default_arg=None)
fixedgrid_dict = unpack_kwarg(kwargs.pop('grid_points', True), default_arg=True)
if isinstance(phases, str):
phases = [phases]
if isinstance(comps, str):
comps = [comps]
if points_dict is None and broadcast is False:
raise ValueError('The \'points\' keyword argument must be specified if broadcast=False is also given.')
components = [x for x in sorted(comps) if not x.startswith('VA')]
# Convert keyword strings to proper state variable objects
# If we don't do this, sympy will get confused during substitution
statevar_dict = collections.OrderedDict((v.StateVariable(key), unpack_condition(value)) \
for (key, value) in sorted(kwargs.items()))
str_statevar_dict = collections.OrderedDict((str(key), unpack_condition(value)) \
for (key, value) in statevar_dict.items())
all_phase_data = []
comp_sets = {}
largest_energy = -np.inf
maximum_internal_dof = 0
# Consider only the active phases
active_phases = dict((name.upper(), dbf.phases[name.upper()]) \
for name in unpack_phases(phases))
for phase_name, phase_obj in sorted(active_phases.items()):
# Build the symbolic representation of the energy
mod = model_dict[phase_name]
# if this is an object type, we need to construct it
if isinstance(mod, type):
try:
model_dict[phase_name] = mod = mod(dbf, comps, phase_name)
except DofError:
# we can't build the specified phase because the
# specified components aren't found in every sublattice
# we'll just skip it
logger.warning("""Suspending specified phase %s due to
some sublattices containing only unspecified components""",
phase_name)
continue
if points_dict[phase_name] is None:
try:
out = getattr(mod, output)
maximum_internal_dof = max(maximum_internal_dof, len(out.atoms(v.SiteFraction)))
except AttributeError:
raise AttributeError('Missing Model attribute {0} specified for {1}'
.format(output, mod.__class__))
else:
maximum_internal_dof = max(maximum_internal_dof, np.asarray(points_dict[phase_name]).shape[-1])
for phase_name, phase_obj in sorted(active_phases.items()):
try:
mod = model_dict[phase_name]
except KeyError:
continue
# Construct an ordered list of the variables
variables, sublattice_dof = generate_dof(phase_obj, mod.components)
# Build the "fast" representation of that model
if callable_dict[phase_name] is None:
out = getattr(mod, output)
# As a last resort, treat undefined symbols as zero
# But warn the user when we do this
# This is consistent with TC's behavior
undefs = list(out.atoms(Symbol) - out.atoms(v.StateVariable))
for undef in undefs:
out = out.xreplace({undef: float(0)})
logger.warning('Setting undefined symbol %s for phase %s to zero',
undef, phase_name)
comp_sets[phase_name] = build_functions(out, list(statevar_dict.keys()) + variables, tmpman=tmpman,
include_obj=True, include_grad=False, include_hess=False)
else:
comp_sets[phase_name] = callable_dict[phase_name]
points = points_dict[phase_name]
if points is None:
# Eliminate pure vacancy endmembers from the calculation
vacancy_indices = list()
for idx, sublattice in enumerate(phase_obj.constituents):
active_in_subl = sorted(set(phase_obj.constituents[idx]).intersection(comps))
if 'VA' in active_in_subl and 'VA' in sorted(comps):
vacancy_indices.append(active_in_subl.index('VA'))
if len(vacancy_indices) != len(phase_obj.constituents):
vacancy_indices = None
logger.debug('vacancy_indices: %s', vacancy_indices)
# Add all endmembers to guarantee their presence
points = endmember_matrix(sublattice_dof,
vacancy_indices=vacancy_indices)
if fixedgrid_dict[phase_name] is True:
# Sample along the edges of the endmembers
# These constitution space edges are often the equilibrium points!
em_pairs = list(itertools.combinations(points, 2))
for first_em, second_em in em_pairs:
extra_points = first_em * np.linspace(0, 1, pdens_dict[phase_name])[np.newaxis].T + \
second_em * np.linspace(0, 1, pdens_dict[phase_name])[::-1][np.newaxis].T
points = np.concatenate((points, extra_points))
# Sample composition space for more points
if sum(sublattice_dof) > len(sublattice_dof):
sampler = sampler_dict[phase_name]
if sampler is None:
sampler = point_sample
points = np.concatenate((points,
sampler(sublattice_dof,
pdof=pdens_dict[phase_name])
))
# If there are nontrivial sublattices with vacancies in them,
# generate a set of points where their fraction is zero and renormalize
for idx, sublattice in enumerate(phase_obj.constituents):
if 'VA' in set(sublattice) and len(sublattice) > 1:
var_idx = variables.index(v.SiteFraction(phase_name, idx, 'VA'))
addtl_pts = np.copy(points)
# set vacancy fraction to log-spaced between 1e-10 and 1e-6
addtl_pts[:, var_idx] = np.power(10.0, -10.0*(1.0 - addtl_pts[:, var_idx]))
# renormalize site fractions
cur_idx = 0
for ctx in sublattice_dof:
end_idx = cur_idx + ctx
addtl_pts[:, cur_idx:end_idx] /= \
addtl_pts[:, cur_idx:end_idx].sum(axis=1)[:, None]
cur_idx = end_idx
# add to points matrix
points = np.concatenate((points, addtl_pts), axis=0)
# Filter out nan's that may have slipped in if we sampled too high a vacancy concentration
# Issues with this appear to be platform-dependent
points = points[~np.isnan(points).any(axis=-1)]
# Ensure that points has the correct dimensions and dtype
points = np.atleast_2d(np.asarray(points, dtype=np.float))
phase_ds = _compute_phase_values(phase_obj, components, variables, str_statevar_dict,
points, comp_sets[phase_name], output,
maximum_internal_dof, broadcast=broadcast)
# largest_energy is really only relevant if fake_points is set
if fake_points:
largest_energy = max(phase_ds[output].max(), largest_energy)
all_phase_data.append(phase_ds)
if fake_points:
if output != 'GM':
raise ValueError('fake_points=True should only be used with output=\'GM\'')
phase_ds = _generate_fake_points(components, statevar_dict, largest_energy, output,
maximum_internal_dof, broadcast)
final_ds = concat(itertools.chain([phase_ds], all_phase_data),
dim='points')
else:
# speedup for single-phase case (found by profiling)
if len(all_phase_data) > 1:
final_ds = concat(all_phase_data, dim='points')
else:
final_ds = all_phase_data[0]
if (not fake_points) and (len(all_phase_data) == 1):
pass
else:
# Reset the points dimension to use a single global index
final_ds['points'] = np.arange(len(final_ds.points))
return final_ds
def equilibrium(dbf,
comps,
phases,
conditions,
output=None,
model=None,
verbose=False,
broadcast=True,
calc_opts=None,
to_xarray=True,
scheduler='sync',
parameters=None,
solver=None,
callables=None,
**kwargs):
"""
Calculate the equilibrium state of a system containing the specified
components and phases, under the specified conditions.
Parameters
----------
dbf : Database
Thermodynamic database containing the relevant parameters.
comps : list
Names of components to consider in the calculation.
phases : list or dict
Names of phases to consider in the calculation.
conditions : dict or (list of dict)
StateVariables and their corresponding value.
output : str or list of str, optional
Additional equilibrium model properties (e.g., CPM, HM, etc.) to compute.
These must be defined as attributes in the Model class of each phase.
model : Model, a dict of phase names to Model, or a seq of both, optional
Model class to use for each phase.
verbose : bool, optional
Print details of calculations. Useful for debugging.
broadcast : bool
If True, broadcast conditions against each other. This will compute all combinations.
If False, each condition should be an equal-length list (or single-valued).
Disabling broadcasting is useful for calculating equilibrium at selected conditions,
when those conditions don't comprise a grid.
calc_opts : dict, optional
Keyword arguments to pass to `calculate`, the energy/property calculation routine.
to_xarray : bool
Whether to return an xarray Dataset (True, default) or an EquilibriumResult.
scheduler : Dask scheduler, optional
Job scheduler for performing the computation.
If None, return a Dask graph of the computation instead of actually doing it.
parameters : dict, optional
Maps SymPy Symbol to numbers, for overriding the values of parameters in the Database.
solver : pycalphad.core.solver.SolverBase
Instance of a solver that is used to calculate local equilibria.
Defaults to a pycalphad.core.solver.InteriorPointSolver.
callables : dict, optional
Pre-computed callable functions for equilibrium calculation.
Returns
-------
Structured equilibrium calculation, or Dask graph if scheduler=None.
Examples
--------
None yet.
"""
if not broadcast:
raise NotImplementedError('Broadcasting cannot yet be disabled')
comps = sorted(unpack_components(dbf, comps))
phases = unpack_phases(phases) or sorted(dbf.phases.keys())
list_of_possible_phases = filter_phases(dbf, comps)
if len(list_of_possible_phases) == 0:
raise ConditionError(
'There are no phases in the Database that can be active with components {0}'
.format(comps))
active_phases = {
name: dbf.phases[name]
for name in filter_phases(dbf, comps, phases)
}
if len(active_phases) == 0:
raise ConditionError(
'None of the passed phases ({0}) are active. List of possible phases: {1}.'
.format(phases, list_of_possible_phases))
if isinstance(comps, (str, v.Species)):
comps = [comps]
if len(set(comps) - set(dbf.species)) > 0:
raise EquilibriumError('Components not found in database: {}'.format(
','.join([c.name for c in (set(comps) - set(dbf.species))])))
calc_opts = calc_opts if calc_opts is not None else dict()
solver = solver if solver is not None else InteriorPointSolver(
verbose=verbose)
parameters = parameters if parameters is not None else dict()
if isinstance(parameters, dict):
parameters = OrderedDict(sorted(parameters.items(), key=str))
models = instantiate_models(dbf,
comps,
active_phases,
model=model,
parameters=parameters)
# Temporary solution until constraint system improves
if conditions.get(v.N) is None:
conditions[v.N] = 1
if np.any(np.array(conditions[v.N]) != 1):
raise ConditionError('N!=1 is not yet supported, got N={}'.format(
conditions[v.N]))
# Modify conditions values to be within numerical limits, e.g., X(AL)=0
# Also wrap single-valued conditions with lists
conds = _adjust_conditions(conditions)
for cond in conds.keys():
if isinstance(cond,
(v.Composition,
v.ChemicalPotential)) and cond.species not in comps:
raise ConditionError(
'{} refers to non-existent component'.format(cond))
state_variables = sorted(get_state_variables(models=models, conds=conds),
key=str)
str_conds = OrderedDict((str(key), value) for key, value in conds.items())
components = [x for x in sorted(comps)]
desired_active_pure_elements = [
list(x.constituents.keys()) for x in components
]
desired_active_pure_elements = [
el.upper() for constituents in desired_active_pure_elements
for el in constituents
]
pure_elements = sorted(
set([x for x in desired_active_pure_elements if x != 'VA']))
if verbose:
print('Components:', ' '.join([str(x) for x in comps]))
print('Phases:', end=' ')
output = output if output is not None else 'GM'
output = output if isinstance(output, (list, tuple, set)) else [output]
output = set(output)
output |= {'GM'}
output = sorted(output)
phase_records = build_phase_records(dbf,
comps,
active_phases,
conds,
models,
output='GM',
callables=callables,
parameters=parameters,
verbose=verbose,
build_gradients=True,
build_hessians=True)
if verbose:
print('[done]', end='\n')
# 'calculate' accepts conditions through its keyword arguments
grid_opts = calc_opts.copy()
statevar_strings = [str(x) for x in state_variables]
grid_opts.update({
key: value
for key, value in str_conds.items() if key in statevar_strings
})
if 'pdens' not in grid_opts:
grid_opts['pdens'] = 500
grid = calculate(dbf,
comps,
active_phases,
model=models,
fake_points=True,
callables=callables,
output='GM',
parameters=parameters,
to_xarray=False,
**grid_opts)
coord_dict = str_conds.copy()
coord_dict['vertex'] = np.arange(
len(pure_elements) + 1
) # +1 is to accommodate the degenerate degree of freedom at the invariant reactions
coord_dict['component'] = pure_elements
properties = starting_point(conds, state_variables, phase_records, grid)
properties = _solve_eq_at_conditions(comps,
properties,
phase_records,
grid,
list(str_conds.keys()),
state_variables,
verbose,
solver=solver)
# Compute equilibrium values of any additional user-specified properties
# We already computed these properties so don't recompute them
output = sorted(set(output) - {'GM', 'MU'})
for out in output:
if (out is None) or (len(out) == 0):
continue
# TODO: How do we know if a specified property should be per_phase or not?
# For now, we make a best guess
if (out == 'degree_of_ordering') or (out == 'DOO'):
per_phase = True
else:
per_phase = False
eqcal = _eqcalculate(dbf,
comps,
active_phases,
conditions,
out,
data=properties,
per_phase=per_phase,
model=models,
callables=callables,
parameters=parameters,
**calc_opts)
properties = properties.merge(eqcal, inplace=True, compat='equals')
if to_xarray:
properties = properties.get_dataset()
properties.attrs['created'] = datetime.utcnow().isoformat()
if len(kwargs) > 0:
warnings.warn(
'The following equilibrium keyword arguments were passed, but unused:\n{}'
.format(kwargs))
return properties
def equilibrium(dbf,
comps,
phases,
conditions,
output=None,
model=None,
verbose=False,
broadcast=True,
calc_opts=None,
scheduler='sync',
parameters=None,
solver=None,
callables=None,
**kwargs):
"""
Calculate the equilibrium state of a system containing the specified
components and phases, under the specified conditions.
Parameters
----------
dbf : Database
Thermodynamic database containing the relevant parameters.
comps : list
Names of components to consider in the calculation.
phases : list or dict
Names of phases to consider in the calculation.
conditions : dict or (list of dict)
StateVariables and their corresponding value.
output : str or list of str, optional
Additional equilibrium model properties (e.g., CPM, HM, etc.) to compute.
These must be defined as attributes in the Model class of each phase.
model : Model, a dict of phase names to Model, or a seq of both, optional
Model class to use for each phase.
verbose : bool, optional
Print details of calculations. Useful for debugging.
broadcast : bool
If True, broadcast conditions against each other. This will compute all combinations.
If False, each condition should be an equal-length list (or single-valued).
Disabling broadcasting is useful for calculating equilibrium at selected conditions,
when those conditions don't comprise a grid.
calc_opts : dict, optional
Keyword arguments to pass to `calculate`, the energy/property calculation routine.
scheduler : Dask scheduler, optional
Job scheduler for performing the computation.
If None, return a Dask graph of the computation instead of actually doing it.
parameters : dict, optional
Maps SymPy Symbol to numbers, for overriding the values of parameters in the Database.
solver : pycalphad.core.solver.SolverBase
Instance of a solver that is used to calculate local equilibria.
Defaults to a pycalphad.core.solver.InteriorPointSolver.
callables : dict, optional
Pre-computed callable functions for equilibrium calculation.
Returns
-------
Structured equilibrium calculation, or Dask graph if scheduler=None.
Examples
--------
None yet.
"""
if not broadcast:
raise NotImplementedError('Broadcasting cannot yet be disabled')
from pycalphad import __version__ as pycalphad_version
comps = sorted(unpack_components(dbf, comps))
phases = unpack_phases(phases) or sorted(dbf.phases.keys())
# remove phases that cannot be active
list_of_possible_phases = filter_phases(dbf, comps)
active_phases = sorted(
set(list_of_possible_phases).intersection(set(phases)))
if len(list_of_possible_phases) == 0:
raise ConditionError(
'There are no phases in the Database that can be active with components {0}'
.format(comps))
if len(active_phases) == 0:
raise ConditionError(
'None of the passed phases ({0}) are active. List of possible phases: {1}.'
.format(phases, list_of_possible_phases))
if isinstance(comps, (str, v.Species)):
comps = [comps]
if len(set(comps) - set(dbf.species)) > 0:
raise EquilibriumError('Components not found in database: {}'.format(
','.join([c.name for c in (set(comps) - set(dbf.species))])))
indep_vars = ['T', 'P']
calc_opts = calc_opts if calc_opts is not None else dict()
model = model if model is not None else Model
solver = solver if solver is not None else InteriorPointSolver(
verbose=verbose)
parameters = parameters if parameters is not None else dict()
if isinstance(parameters, dict):
parameters = OrderedDict(sorted(parameters.items(), key=str))
# Modify conditions values to be within numerical limits, e.g., X(AL)=0
# Also wrap single-valued conditions with lists
conds = _adjust_conditions(conditions)
for cond in conds.keys():
if isinstance(cond,
(v.Composition,
v.ChemicalPotential)) and cond.species not in comps:
raise ConditionError(
'{} refers to non-existent component'.format(cond))
str_conds = OrderedDict((str(key), value) for key, value in conds.items())
num_calcs = np.prod([len(i) for i in str_conds.values()])
components = [x for x in sorted(comps)]
desired_active_pure_elements = [
list(x.constituents.keys()) for x in components
]
desired_active_pure_elements = [
el.upper() for constituents in desired_active_pure_elements
for el in constituents
]
pure_elements = sorted(
set([x for x in desired_active_pure_elements if x != 'VA']))
other_output_callables = {}
if verbose:
print('Components:', ' '.join([str(x) for x in comps]))
print('Phases:', end=' ')
output = output if output is not None else 'GM'
output = output if isinstance(output, (list, tuple, set)) else [output]
output = set(output)
output |= {'GM'}
output = sorted(output)
for o in output:
if o == 'GM':
eq_callables = build_callables(dbf,
comps,
active_phases,
model=model,
parameters=parameters,
output=o,
build_gradients=True,
callables=callables,
verbose=verbose)
else:
other_output_callables[o] = build_callables(dbf,
comps,
active_phases,
model=model,
parameters=parameters,
output=o,
build_gradients=False,
verbose=False)
phase_records = eq_callables['phase_records']
models = eq_callables['model']
maximum_internal_dof = max(
len(mod.site_fractions) for mod in models.values())
if verbose:
print('[done]', end='\n')
# 'calculate' accepts conditions through its keyword arguments
grid_opts = calc_opts.copy()
grid_opts.update(
{key: value
for key, value in str_conds.items() if key in indep_vars})
if 'pdens' not in grid_opts:
grid_opts['pdens'] = 500
coord_dict = str_conds.copy()
coord_dict['vertex'] = np.arange(
len(pure_elements) + 1
) # +1 is to accommodate the degenerate degree of freedom at the invariant reactions
grid_shape = np.meshgrid(*coord_dict.values(), indexing='ij',
sparse=False)[0].shape
coord_dict['component'] = pure_elements
grid = delayed(calculate, pure=False)(dbf,
comps,
active_phases,
output='GM',
model=models,
fake_points=True,
callables=eq_callables,
parameters=parameters,
**grid_opts)
max_phase_name_len = max(len(name) for name in active_phases)
# Need to allow for '_FAKE_' psuedo-phase
max_phase_name_len = max(max_phase_name_len, 6)
properties = delayed(Dataset, pure=False)(
{
'NP': (list(str_conds.keys()) + ['vertex'], np.empty(grid_shape)),
'GM': (list(str_conds.keys()), np.empty(grid_shape[:-1])),
'MU': (list(str_conds.keys()) + ['component'],
np.empty(grid_shape[:-1] + (len(pure_elements), ))),
'X': (list(str_conds.keys()) + ['vertex', 'component'],
np.empty(grid_shape + (len(pure_elements), ))),
'Y': (list(str_conds.keys()) + ['vertex', 'internal_dof'],
np.empty(grid_shape + (maximum_internal_dof, ))),
'Phase': (list(str_conds.keys()) + ['vertex'],
np.empty(grid_shape, dtype='U%s' % max_phase_name_len)),
'points': (list(str_conds.keys()) + ['vertex'],
np.empty(grid_shape, dtype=np.int32))
},
coords=coord_dict,
attrs={
'engine': 'pycalphad %s' % pycalphad_version
},
)
# One last call to ensure 'properties' and 'grid' are consistent with one another
properties = delayed(lower_convex_hull, pure=False)(grid, properties)
conditions_per_chunk_per_axis = 2
if num_calcs > 1:
# Generate slices of 'properties'
slices = []
for val in grid_shape[:-1]:
idx_arr = list(range(val))
num_chunks = int(np.floor(val / conditions_per_chunk_per_axis))
if num_chunks > 0:
cond_slices = [
x for x in np.array_split(np.asarray(idx_arr), num_chunks)
if len(x) > 0
]
else:
cond_slices = [idx_arr]
slices.append(cond_slices)
chunk_dims = [len(slc) for slc in slices]
chunk_grid = np.array(
np.unravel_index(np.arange(np.prod(chunk_dims)), chunk_dims)).T
res = []
for chunk in chunk_grid:
prop_slice = properties[OrderedDict(
list(
zip(str_conds.keys(), [
np.atleast_1d(sl)[ch] for ch, sl in zip(chunk, slices)
])))]
job = delayed(_solve_eq_at_conditions,
pure=False)(comps,
prop_slice,
phase_records,
grid,
list(str_conds.keys()),
verbose,
solver=solver)
res.append(job)
properties = delayed(_merge_property_slices,
pure=False)(properties, chunk_grid, slices,
list(str_conds.keys()), res)
else:
# Single-process job; don't create child processes
properties = delayed(_solve_eq_at_conditions,
pure=False)(comps,
properties,
phase_records,
grid,
list(str_conds.keys()),
verbose,
solver=solver)
# Compute equilibrium values of any additional user-specified properties
# We already computed these properties so don't recompute them
output = sorted(set(output) - {'GM', 'MU'})
for out in output:
if (out is None) or (len(out) == 0):
continue
# TODO: How do we know if a specified property should be per_phase or not?
# For now, we make a best guess
if (out == 'degree_of_ordering') or (out == 'DOO'):
per_phase = True
else:
per_phase = False
eqcal = delayed(_eqcalculate,
pure=False)(dbf,
comps,
active_phases,
conditions,
out,
data=properties,
per_phase=per_phase,
callables=other_output_callables[out],
parameters=parameters,
model=models,
**calc_opts)
properties = delayed(properties.merge, pure=False)(eqcal,
inplace=True,
compat='equals')
if scheduler is not None:
properties = dask.compute(properties, scheduler=scheduler)[0]
properties.attrs['created'] = datetime.utcnow().isoformat()
if len(kwargs) > 0:
warnings.warn(
'The following equilibrium keyword arguments were passed, but unused:\n{}'
.format(kwargs))
return properties
def _eqcalculate(dbf,
comps,
phases,
conditions,
output,
data=None,
per_phase=False,
callables=None,
model=None,
parameters=None,
**kwargs):
"""
WARNING: API/calling convention not finalized.
Compute the *equilibrium value* of a property.
This function differs from `calculate` in that it computes
thermodynamic equilibrium instead of randomly sampling the
internal degrees of freedom of a phase.
Because of that, it's slower than `calculate`.
This plugs in the equilibrium phase and site fractions
to compute a thermodynamic property defined in a Model.
Parameters
----------
dbf : Database
Thermodynamic database containing the relevant parameters.
comps : list
Names of components to consider in the calculation.
phases : list or dict
Names of phases to consider in the calculation.
conditions : dict or (list of dict)
StateVariables and their corresponding value.
output : str
Equilibrium model property (e.g., CPM, HM, etc.) to compute.
This must be defined as an attribute in the Model class of each phase.
data : Dataset
Previous result of call to `equilibrium`.
Should contain the equilibrium configurations at the conditions of interest.
If the databases are not the same as in the original calculation,
the results may be meaningless.
per_phase : bool, optional
If True, compute and return the property for each phase present.
If False, return the total system value, weighted by the phase fractions.
callables : dict
Callable functions to compute 'output' for each phase.
model : a dict of phase names to Model
Model class to use for each phase.
parameters : dict, optional
Maps SymPy Symbol to numbers, for overriding the values of parameters in the Database.
kwargs
Passed to `calculate`.
Returns
-------
Dataset of property as a function of equilibrium conditions
"""
if data is None:
raise ValueError('Required kwarg "data" is not specified')
if model is None:
raise ValueError('Required kwarg "model" is not specified')
active_phases = unpack_phases(phases)
conds = _adjust_conditions(conditions)
indep_vars = ['N', 'P', 'T']
# TODO: Rewrite this to use the coord dict from 'data'
str_conds = OrderedDict((str(key), value) for key, value in conds.items())
indep_vals = list([float(x) for x in np.atleast_1d(val)]
for key, val in str_conds.items() if key in indep_vars)
coord_dict = str_conds.copy()
components = [x for x in sorted(comps)]
desired_active_pure_elements = [
list(x.constituents.keys()) for x in components
]
desired_active_pure_elements = [
el.upper() for constituents in desired_active_pure_elements
for el in constituents
]
pure_elements = sorted(
set([x for x in desired_active_pure_elements if x != 'VA']))
coord_dict['vertex'] = np.arange(
len(pure_elements) + 1
) # +1 is to accommodate the degenerate degree of freedom at the invariant reactions
grid_shape = np.meshgrid(*coord_dict.values(), indexing='ij',
sparse=False)[0].shape
prop_shape = grid_shape
prop_dims = list(str_conds.keys()) + ['vertex']
result = LightDataset({output: (prop_dims, np.full(prop_shape, np.nan))},
coords=coord_dict)
# For each phase select all conditions where that phase exists
# Perform the appropriate calculation and then write the result back
for phase in active_phases:
dof = len(model[phase].site_fractions)
current_phase_indices = (data.Phase == phase)
if ~np.any(current_phase_indices):
continue
points = data.Y[np.nonzero(current_phase_indices)][..., :dof]
statevar_indices = np.nonzero(current_phase_indices)[:len(indep_vals)]
statevars = {
key: np.take(np.asarray(vals), idx)
for key, vals, idx in zip(indep_vars, indep_vals, statevar_indices)
}
statevars.update(kwargs)
if statevars.get('mode', None) is None:
statevars['mode'] = 'numpy'
calcres = calculate(dbf,
comps, [phase],
output=output,
points=points,
broadcast=False,
callables=callables,
parameters=parameters,
model=model,
**statevars)
result[output][np.nonzero(
current_phase_indices)] = calcres[output].values
if not per_phase:
out = np.nansum(result[output] * data['NP'], axis=-1)
dv_output = result.data_vars[output]
result.remove(output)
# remove the vertex coordinate because we summed over it
result.add_variable(output, dv_output[0][:-1], out)
else:
dv_phase = data.data_vars['Phase']
dv_np = data.data_vars['NP']
result.add_variable('Phase', dv_phase[0], dv_phase[1])
result.add_variable('NP', dv_np[0], dv_np[1])
return result
def equilibrium(dbf, comps, phases, conditions, output=None, model=None,
verbose=False, broadcast=True, calc_opts=None,
scheduler=dask.local.get_sync,
parameters=None, **kwargs):
"""
Calculate the equilibrium state of a system containing the specified
components and phases, under the specified conditions.
Parameters
----------
dbf : Database
Thermodynamic database containing the relevant parameters.
comps : list
Names of components to consider in the calculation.
phases : list or dict
Names of phases to consider in the calculation.
conditions : dict or (list of dict)
StateVariables and their corresponding value.
output : str or list of str, optional
Additional equilibrium model properties (e.g., CPM, HM, etc.) to compute.
These must be defined as attributes in the Model class of each phase.
model : Model, a dict of phase names to Model, or a seq of both, optional
Model class to use for each phase.
verbose : bool, optional
Print details of calculations. Useful for debugging.
broadcast : bool
If True, broadcast conditions against each other. This will compute all combinations.
If False, each condition should be an equal-length list (or single-valued).
Disabling broadcasting is useful for calculating equilibrium at selected conditions,
when those conditions don't comprise a grid.
calc_opts : dict, optional
Keyword arguments to pass to `calculate`, the energy/property calculation routine.
scheduler : Dask scheduler, optional
Job scheduler for performing the computation.
If None, return a Dask graph of the computation instead of actually doing it.
parameters : dict, optional
Maps SymPy Symbol to numbers, for overriding the values of parameters in the Database.
Returns
-------
Structured equilibrium calculation, or Dask graph if scheduler=None.
Examples
--------
None yet.
"""
if not broadcast:
raise NotImplementedError('Broadcasting cannot yet be disabled')
from pycalphad import __version__ as pycalphad_version
comps = sorted(unpack_components(dbf, comps))
phases = unpack_phases(phases) or sorted(dbf.phases.keys())
# remove phases that cannot be active
list_of_possible_phases = filter_phases(dbf, comps)
active_phases = sorted(set(list_of_possible_phases).intersection(set(phases)))
if len(list_of_possible_phases) == 0:
raise ConditionError('There are no phases in the Database that can be active with components {0}'.format(comps))
if len(active_phases) == 0:
raise ConditionError('None of the passed phases ({0}) are active. List of possible phases: {1}.'.format(phases, list_of_possible_phases))
if isinstance(comps, (str, v.Species)):
comps = [comps]
if len(set(comps) - set(dbf.species)) > 0:
raise EquilibriumError('Components not found in database: {}'
.format(','.join([c.name for c in (set(comps) - set(dbf.species))])))
indep_vars = ['T', 'P']
calc_opts = calc_opts if calc_opts is not None else dict()
model = model if model is not None else Model
phase_records = dict()
diagnostic = kwargs.pop('_diagnostic', False)
callable_dict = kwargs.pop('callables', dict())
mass_dict = unpack_kwarg(kwargs.pop('massfuncs', None), default_arg=None)
mass_grad_dict = unpack_kwarg(kwargs.pop('massgradfuncs', None), default_arg=None)
grad_callable_dict = kwargs.pop('grad_callables', dict())
hess_callable_dict = kwargs.pop('hess_callables', dict())
parameters = parameters if parameters is not None else dict()
if isinstance(parameters, dict):
parameters = OrderedDict(sorted(parameters.items(), key=str))
param_symbols = tuple(parameters.keys())
param_values = np.atleast_1d(np.array(list(parameters.values()), dtype=np.float))
maximum_internal_dof = 0
# Modify conditions values to be within numerical limits, e.g., X(AL)=0
# Also wrap single-valued conditions with lists
conds = _adjust_conditions(conditions)
for cond in conds.keys():
if isinstance(cond, (v.Composition, v.ChemicalPotential)) and cond.species not in comps:
raise ConditionError('{} refers to non-existent component'.format(cond))
str_conds = OrderedDict((str(key), value) for key, value in conds.items())
num_calcs = np.prod([len(i) for i in str_conds.values()])
indep_vals = list([float(x) for x in np.atleast_1d(val)]
for key, val in str_conds.items() if key in indep_vars)
components = [x for x in sorted(comps)]
desired_active_pure_elements = [list(x.constituents.keys()) for x in components]
desired_active_pure_elements = [el.upper() for constituents in desired_active_pure_elements for el in constituents]
pure_elements = sorted(set([x for x in desired_active_pure_elements if x != 'VA']))
# Construct models for each phase; prioritize user models
models = unpack_kwarg(model, default_arg=Model)
if verbose:
print('Components:', ' '.join([str(x) for x in comps]))
print('Phases:', end=' ')
max_phase_name_len = max(len(name) for name in active_phases)
# Need to allow for '_FAKE_' psuedo-phase
max_phase_name_len = max(max_phase_name_len, 6)
for name in active_phases:
mod = models[name]
if isinstance(mod, type):
models[name] = mod = mod(dbf, comps, name, parameters=parameters)
site_fracs = mod.site_fractions
variables = sorted(site_fracs, key=str)
maximum_internal_dof = max(maximum_internal_dof, len(site_fracs))
out = models[name].energy
if (not callable_dict.get(name, False)) or not (grad_callable_dict.get(name, False)):
# Only force undefineds to zero if we're not overriding them
undefs = [x for x in out.free_symbols if (not isinstance(x, v.StateVariable)) and not (x in param_symbols)]
for undef in undefs:
out = out.xreplace({undef: float(0)})
cf, gf = build_functions(out, tuple([v.P, v.T] + site_fracs),
parameters=param_symbols)
hf = None
if callable_dict.get(name, None) is None:
callable_dict[name] = cf
if grad_callable_dict.get(name, None) is None:
grad_callable_dict[name] = gf
if hess_callable_dict.get(name, None) is None:
hess_callable_dict[name] = hf
if (mass_dict[name] is None) or (mass_grad_dict[name] is None):
# TODO: In principle, we should also check for undefs in mod.moles()
tup1, tup2 = zip(*[build_functions(mod.moles(el), [v.P, v.T] + variables,
include_obj=True, include_grad=True,
parameters=param_symbols)
for el in pure_elements])
if mass_dict[name] is None:
mass_dict[name] = tup1
if mass_grad_dict[name] is None:
mass_grad_dict[name] = tup2
phase_records[name.upper()] = PhaseRecord_from_cython(comps, variables,
np.array(dbf.phases[name].sublattices, dtype=np.float),
param_values, callable_dict[name],
grad_callable_dict[name], hess_callable_dict[name],
mass_dict[name], mass_grad_dict[name])
if verbose:
print(name, end=' ')
if verbose:
print('[done]', end='\n')
# 'calculate' accepts conditions through its keyword arguments
grid_opts = calc_opts.copy()
grid_opts.update({key: value for key, value in str_conds.items() if key in indep_vars})
if 'pdens' not in grid_opts:
grid_opts['pdens'] = 500
coord_dict = str_conds.copy()
coord_dict['vertex'] = np.arange(len(pure_elements))
grid_shape = np.meshgrid(*coord_dict.values(),
indexing='ij', sparse=False)[0].shape
coord_dict['component'] = pure_elements
grid = delayed(calculate, pure=False)(dbf, comps, active_phases, output='GM',
model=models, callables=callable_dict, massfuncs=mass_dict,
fake_points=True, parameters=parameters, **grid_opts)
properties = delayed(Dataset, pure=False)({'NP': (list(str_conds.keys()) + ['vertex'],
np.empty(grid_shape)),
'GM': (list(str_conds.keys()),
np.empty(grid_shape[:-1])),
'MU': (list(str_conds.keys()) + ['component'],
np.empty(grid_shape)),
'X': (list(str_conds.keys()) + ['vertex', 'component'],
np.empty(grid_shape + (grid_shape[-1],))),
'Y': (list(str_conds.keys()) + ['vertex', 'internal_dof'],
np.empty(grid_shape + (maximum_internal_dof,))),
'Phase': (list(str_conds.keys()) + ['vertex'],
np.empty(grid_shape, dtype='U%s' % max_phase_name_len)),
'points': (list(str_conds.keys()) + ['vertex'],
np.empty(grid_shape, dtype=np.int32))
},
coords=coord_dict,
attrs={'engine': 'pycalphad %s' % pycalphad_version},
)
# One last call to ensure 'properties' and 'grid' are consistent with one another
properties = delayed(lower_convex_hull, pure=False)(grid, properties)
conditions_per_chunk_per_axis = 2
if num_calcs > 1:
# Generate slices of 'properties'
slices = []
for val in grid_shape[:-1]:
idx_arr = list(range(val))
num_chunks = int(np.floor(val/conditions_per_chunk_per_axis))
if num_chunks > 0:
cond_slices = [x for x in np.array_split(np.asarray(idx_arr), num_chunks) if len(x) > 0]
else:
cond_slices = [idx_arr]
slices.append(cond_slices)
chunk_dims = [len(slc) for slc in slices]
chunk_grid = np.array(np.unravel_index(np.arange(np.prod(chunk_dims)), chunk_dims)).T
res = []
for chunk in chunk_grid:
prop_slice = properties[OrderedDict(list(zip(str_conds.keys(),
[np.atleast_1d(sl)[ch] for ch, sl in zip(chunk, slices)])))]
job = delayed(_solve_eq_at_conditions, pure=False)(comps, prop_slice, phase_records, grid,
list(str_conds.keys()), verbose)
res.append(job)
properties = delayed(_merge_property_slices, pure=False)(properties, chunk_grid, slices, list(str_conds.keys()), res)
else:
# Single-process job; don't create child processes
properties = delayed(_solve_eq_at_conditions, pure=False)(comps, properties, phase_records, grid,
list(str_conds.keys()), verbose)
# Compute equilibrium values of any additional user-specified properties
output = output if isinstance(output, (list, tuple, set)) else [output]
# We already computed these properties so don't recompute them
output = sorted(set(output) - {'GM', 'MU'})
for out in output:
if (out is None) or (len(out) == 0):
continue
# TODO: How do we know if a specified property should be per_phase or not?
# For now, we make a best guess
if (out == 'degree_of_ordering') or (out == 'DOO'):
per_phase = True
else:
per_phase = False
eqcal = delayed(_eqcalculate, pure=False)(dbf, comps, active_phases, conditions, out,
data=properties, per_phase=per_phase, model=models, **calc_opts)
properties = delayed(properties.merge, pure=False)(eqcal, inplace=True, compat='equals')
if scheduler is not None:
properties = dask.compute(properties, get=scheduler)[0]
properties.attrs['created'] = datetime.utcnow().isoformat()
if len(kwargs) > 0:
warnings.warn('The following equilibrium keyword arguments were passed, but unused:\n{}'.format(kwargs))
return properties
def equilibrium(dbf, comps, phases, conditions, **kwargs):
"""
Calculate the equilibrium state of a system containing the specified
components and phases, under the specified conditions.
Model parameters are taken from 'dbf'.
Parameters
----------
dbf : Database
Thermodynamic database containing the relevant parameters.
comps : list
Names of components to consider in the calculation.
phases : list or dict
Names of phases to consider in the calculation.
conditions : dict or (list of dict)
StateVariables and their corresponding value.
verbose : bool, optional (Default: True)
Show progress of calculations.
grid_opts : dict, optional
Keyword arguments to pass to the initial grid routine.
Returns
-------
Structured equilibrium calculation.
Examples
--------
None yet.
"""
active_phases = unpack_phases(phases) or sorted(dbf.phases.keys())
comps = sorted(comps)
indep_vars = ['T', 'P']
grid_opts = kwargs.pop('grid_opts', dict())
verbose = kwargs.pop('verbose', True)
phase_records = dict()
callable_dict = kwargs.pop('callables', dict())
grad_callable_dict = kwargs.pop('grad_callables', dict())
hess_callable_dict = kwargs.pop('hess_callables', dict())
points_dict = dict()
maximum_internal_dof = 0
conds = OrderedDict((key, unpack_condition(value)) for key, value in sorted(conditions.items(), key=str))
str_conds = OrderedDict((str(key), value) for key, value in conds.items())
indep_vals = list([float(x) for x in np.atleast_1d(val)]
for key, val in str_conds.items() if key in indep_vars)
components = [x for x in sorted(comps) if not x.startswith('VA')]
# Construct models for each phase; prioritize user models
models = unpack_kwarg(kwargs.pop('model', Model), default_arg=Model)
if verbose:
print('Components:', ' '.join(comps))
print('Phases:', end=' ')
for name in active_phases:
mod = models[name]
if isinstance(mod, type):
models[name] = mod = mod(dbf, comps, name)
variables = sorted(mod.energy.atoms(v.StateVariable).union({key for key in conditions.keys() if key in [v.T, v.P]}), key=str)
site_fracs = sorted(mod.energy.atoms(v.SiteFraction), key=str)
maximum_internal_dof = max(maximum_internal_dof, len(site_fracs))
# Extra factor '1e-100...' is to work around an annoying broadcasting bug for zero gradient entries
#models[name].models['_broadcaster'] = 1e-100 * Mul(*variables) ** 3
out = models[name].energy
undefs = list(out.atoms(Symbol) - out.atoms(v.StateVariable))
for undef in undefs:
out = out.xreplace({undef: float(0)})
callable_dict[name], grad_callable_dict[name], hess_callable_dict[name] = \
build_functions(out, [v.P, v.T] + site_fracs)
# Adjust gradient by the approximate chemical potentials
hyperplane = Add(*[v.MU(i)*mole_fraction(dbf.phases[name], comps, i)
for i in comps if i != 'VA'])
plane_obj, plane_grad, plane_hess = build_functions(hyperplane, [v.MU(i) for i in comps if i != 'VA']+site_fracs)
phase_records[name.upper()] = PhaseRecord(variables=variables,
grad=grad_callable_dict[name],
hess=hess_callable_dict[name],
plane_grad=plane_grad,
plane_hess=plane_hess)
if verbose:
print(name, end=' ')
if verbose:
print('[done]', end='\n')
# 'calculate' accepts conditions through its keyword arguments
grid_opts.update({key: value for key, value in str_conds.items() if key in indep_vars})
if 'pdens' not in grid_opts:
grid_opts['pdens'] = 100
coord_dict = str_conds.copy()
coord_dict['vertex'] = np.arange(len(components))
grid_shape = np.meshgrid(*coord_dict.values(),
indexing='ij', sparse=False)[0].shape
coord_dict['component'] = components
if verbose:
print('Computing initial grid', end=' ')
grid = calculate(dbf, comps, active_phases, output='GM',
model=models, callables=callable_dict, fake_points=True, **grid_opts)
if verbose:
print('[{0} points, {1}]'.format(len(grid.points), sizeof_fmt(grid.nbytes)), end='\n')
properties = xray.Dataset({'NP': (list(str_conds.keys()) + ['vertex'],
np.empty(grid_shape)),
'GM': (list(str_conds.keys()),
np.empty(grid_shape[:-1])),
'MU': (list(str_conds.keys()) + ['component'],
np.empty(grid_shape)),
'points': (list(str_conds.keys()) + ['vertex'],
np.empty(grid_shape, dtype=np.int))
},
coords=coord_dict,
attrs={'iterations': 1},
)
# Store the potentials from the previous iteration
current_potentials = properties.MU.copy()
for iteration in range(MAX_ITERATIONS):
if verbose:
print('Computing convex hull [iteration {}]'.format(properties.attrs['iterations']))
# lower_convex_hull will modify properties
lower_convex_hull(grid, properties)
progress = np.abs(current_potentials - properties.MU).values
converged = (progress < MIN_PROGRESS).all(axis=-1)
if verbose:
print('progress', progress.max(), '[{} conditions updated]'.format(np.sum(~converged)))
if progress.max() < MIN_PROGRESS:
if verbose:
print('Convergence achieved')
break
current_potentials[...] = properties.MU.values
if verbose:
print('Refining convex hull')
# Insert extra dimensions for non-T,P conditions so GM broadcasts correctly
energy_broadcast_shape = grid.GM.values.shape[:len(indep_vals)] + \
(1,) * (len(str_conds) - len(indep_vals)) + (grid.GM.values.shape[-1],)
driving_forces = np.einsum('...i,...i',
properties.MU.values[..., np.newaxis, :].astype(np.float),
grid.X.values[np.index_exp[...] +
(np.newaxis,) * (len(str_conds) - len(indep_vals)) +
np.index_exp[:, :]].astype(np.float)) - \
grid.GM.values.view().reshape(energy_broadcast_shape)
for name in active_phases:
dof = len(models[name].energy.atoms(v.SiteFraction))
current_phase_indices = (grid.Phase.values == name).reshape(energy_broadcast_shape[:-1] + (-1,))
# Broadcast to capture all conditions
current_phase_indices = np.broadcast_arrays(current_phase_indices,
np.empty(driving_forces.shape))[0]
# This reshape is safe as long as phases have the same number of points at all indep. conditions
current_phase_driving_forces = driving_forces[current_phase_indices].reshape(
current_phase_indices.shape[:-1] + (-1,))
# Note: This works as long as all points are in the same phase order for all T, P
current_site_fractions = grid.Y.values[..., current_phase_indices[(0,) * len(str_conds)], :]
if np.sum(current_site_fractions[(0,) * len(indep_vals)][..., :dof]) == dof:
# All site fractions are 1, aka zero internal degrees of freedom
# Impossible to refine these points, so skip this phase
points_dict[name] = current_site_fractions[(0,) * len(indep_vals)][..., :dof]
continue
# Find the N points with largest driving force for a given set of conditions
# Remember that driving force has a sign, so we want the "most positive" values
# N is the number of components, in this context
# N points define a 'best simplex' for every set of conditions
# We also need to restrict ourselves to one phase at a time
trial_indices = np.argpartition(current_phase_driving_forces,
-len(components), axis=-1)[..., -len(components):]
trial_indices = trial_indices.ravel()
statevar_indices = np.unravel_index(np.arange(np.multiply.reduce(properties.GM.values.shape + (len(components),))),
properties.GM.values.shape + (len(components),))[:len(indep_vals)]
points = current_site_fractions[np.index_exp[statevar_indices + (trial_indices,)]]
points.shape = properties.points.shape[:-1] + (-1, maximum_internal_dof)
# The Y arrays have been padded, so we should slice off the padding
points = points[..., :dof]
#print('Starting points shape: ', points.shape)
#print(points)
if len(points) == 0:
if name in points_dict:
del points_dict[name]
# No nearly stable points: skip this phase
continue
num_vars = len(phase_records[name].variables)
plane_grad = phase_records[name].plane_grad
plane_hess = phase_records[name].plane_hess
statevar_grid = np.meshgrid(*itertools.chain(indep_vals), sparse=True, indexing='ij')
# TODO: A more sophisticated treatment of constraints
num_constraints = len(dbf.phases[name].sublattices)
constraint_jac = np.zeros((num_constraints, num_vars-len(indep_vars)))
# Independent variables are always fixed (in this limited implementation)
#for idx in range(len(indep_vals)):
# constraint_jac[idx, idx] = 1
# This is for site fraction balance constraints
var_idx = 0#len(indep_vals)
for idx in range(len(dbf.phases[name].sublattices)):
active_in_subl = set(dbf.phases[name].constituents[idx]).intersection(comps)
constraint_jac[idx,
var_idx:var_idx + len(active_in_subl)] = 1
var_idx += len(active_in_subl)
newton_iteration = 0
while newton_iteration < MAX_NEWTON_ITERATIONS:
flattened_points = points.reshape(points.shape[:len(indep_vals)] + (-1, points.shape[-1]))
grad_args = itertools.chain([i[..., None] for i in statevar_grid],
[flattened_points[..., i] for i in range(flattened_points.shape[-1])])
grad = np.array(phase_records[name].grad(*grad_args), dtype=np.float)
# Remove derivatives wrt T,P
grad = grad[..., len(indep_vars):]
grad.shape = points.shape
grad[np.isnan(grad).any(axis=-1)] = 0 # This is necessary for gradients on the edge of space
hess_args = itertools.chain([i[..., None] for i in statevar_grid],
[flattened_points[..., i] for i in range(flattened_points.shape[-1])])
hess = np.array(phase_records[name].hess(*hess_args), dtype=np.float)
# Remove derivatives wrt T,P
hess = hess[..., len(indep_vars):, len(indep_vars):]
hess.shape = points.shape + (hess.shape[-1],)
hess[np.isnan(hess).any(axis=(-2, -1))] = np.eye(hess.shape[-1])
plane_args = itertools.chain([properties.MU.values[..., i][..., None] for i in range(properties.MU.shape[-1])],
[points[..., i] for i in range(points.shape[-1])])
cast_grad = np.array(plane_grad(*plane_args), dtype=np.float)
# Remove derivatives wrt chemical potentials
cast_grad = cast_grad[..., properties.MU.shape[-1]:]
grad = grad - cast_grad
plane_args = itertools.chain([properties.MU.values[..., i][..., None] for i in range(properties.MU.shape[-1])],
[points[..., i] for i in range(points.shape[-1])])
cast_hess = np.array(plane_hess(*plane_args), dtype=np.float)
# Remove derivatives wrt chemical potentials
cast_hess = cast_hess[..., properties.MU.shape[-1]:, properties.MU.shape[-1]:]
cast_hess = -cast_hess + hess
hess = cast_hess.astype(np.float, copy=False)
try:
e_matrix = np.linalg.inv(hess)
except np.linalg.LinAlgError:
print(hess)
raise
current = calculate(dbf, comps, name, output='GM',
model=models, callables=callable_dict,
fake_points=False,
points=points.reshape(points.shape[:len(indep_vals)] + (-1, points.shape[-1])),
**grid_opts)
current_plane = np.multiply(current.X.values.reshape(points.shape[:-1] + (len(components),)),
properties.MU.values[..., np.newaxis, :]).sum(axis=-1)
current_df = current.GM.values.reshape(points.shape[:-1]) - current_plane
#print('Inv hess check: ', np.isnan(e_matrix).any())
#print('grad check: ', np.isnan(grad).any())
dy_unconstrained = -np.einsum('...ij,...j->...i', e_matrix, grad)
#print('dy_unconstrained check: ', np.isnan(dy_unconstrained).any())
proj_matrix = np.dot(e_matrix, constraint_jac.T)
inv_matrix = np.rollaxis(np.dot(constraint_jac, proj_matrix), 0, -1)
inv_term = np.linalg.inv(inv_matrix)
#print('inv_term check: ', np.isnan(inv_term).any())
first_term = np.einsum('...ij,...jk->...ik', proj_matrix, inv_term)
#print('first_term check: ', np.isnan(first_term).any())
# Normally a term for the residual here
# We only choose starting points which obey the constraints, so r = 0
cons_summation = np.einsum('...i,...ji->...j', dy_unconstrained, constraint_jac)
#print('cons_summation check: ', np.isnan(cons_summation).any())
cons_correction = np.einsum('...ij,...j->...i', first_term, cons_summation)
#print('cons_correction check: ', np.isnan(cons_correction).any())
dy_constrained = dy_unconstrained - cons_correction
#print('dy_constrained check: ', np.isnan(dy_constrained).any())
# TODO: Support for adaptive changing independent variable steps
new_direction = dy_constrained
#print('new_direction', new_direction)
#print('points', points)
# Backtracking line search
if np.isnan(new_direction).any():
print('new_direction', new_direction)
#print('Convergence angle:', -(grad*new_direction).sum(axis=-1) / (np.linalg.norm(grad, axis=-1) * np.linalg.norm(new_direction, axis=-1)))
new_points = points + INITIAL_STEP_SIZE * new_direction
alpha = np.full(new_points.shape[:-1], INITIAL_STEP_SIZE, dtype=np.float)
alpha[np.all(np.linalg.norm(new_direction, axis=-1) < MIN_DIRECTION_NORM, axis=-1)] = 0
negative_points = np.any(new_points < 0., axis=-1)
while np.any(negative_points):
alpha[negative_points] *= 0.5
new_points = points + alpha[..., np.newaxis] * new_direction
negative_points = np.any(new_points < 0., axis=-1)
# Backtracking line search
# alpha now contains maximum possible values that keep us inside the space
# but we don't just want to take the biggest step; we want the biggest step which reduces energy
new_points = new_points.reshape(new_points.shape[:len(indep_vals)] + (-1, new_points.shape[-1]))
candidates = calculate(dbf, comps, name, output='GM',
model=models, callables=callable_dict,
fake_points=False, points=new_points, **grid_opts)
candidate_plane = np.multiply(candidates.X.values.reshape(points.shape[:-1] + (len(components),)),
properties.MU.values[..., np.newaxis, :]).sum(axis=-1)
energy_diff = (candidates.GM.values.reshape(new_direction.shape[:-1]) - candidate_plane) - current_df
new_points.shape = new_direction.shape
bad_steps = energy_diff > alpha * 1e-4 * (new_direction * grad).sum(axis=-1)
backtracking_iterations = 0
while np.any(bad_steps):
alpha[bad_steps] *= 0.5
new_points = points + alpha[..., np.newaxis] * new_direction
#print('new_points', new_points)
#print('bad_steps', bad_steps)
new_points = new_points.reshape(new_points.shape[:len(indep_vals)] + (-1, new_points.shape[-1]))
candidates = calculate(dbf, comps, name, output='GM',
model=models, callables=callable_dict,
fake_points=False, points=new_points, **grid_opts)
candidate_plane = np.multiply(candidates.X.values.reshape(points.shape[:-1] + (len(components),)),
properties.MU.values[..., np.newaxis, :]).sum(axis=-1)
energy_diff = (candidates.GM.values.reshape(new_direction.shape[:-1]) - candidate_plane) - current_df
#print('energy_diff', energy_diff)
new_points.shape = new_direction.shape
bad_steps = energy_diff > alpha * 1e-4 * (new_direction * grad).sum(axis=-1)
backtracking_iterations += 1
if backtracking_iterations > MAX_BACKTRACKING:
break
biggest_step = np.max(np.linalg.norm(new_points - points, axis=-1))
if biggest_step < 1e-2:
if verbose:
print('N-R convergence on mini-iteration', newton_iteration, '[{}]'.format(name))
points = new_points
break
if verbose:
#print('Biggest step:', biggest_step)
#print('points', points)
#print('grad of points', grad)
#print('new_direction', new_direction)
#print('alpha', alpha)
#print('new_points', new_points)
pass
points = new_points
newton_iteration += 1
new_points = points.reshape(points.shape[:len(indep_vals)] + (-1, points.shape[-1]))
new_points = np.concatenate((current_site_fractions[..., :dof], new_points), axis=-2)
points_dict[name] = new_points
if verbose:
print('Rebuilding grid', end=' ')
grid = calculate(dbf, comps, active_phases, output='GM',
model=models, callables=callable_dict,
fake_points=True, points=points_dict, **grid_opts)
if verbose:
print('[{0} points, {1}]'.format(len(grid.points), sizeof_fmt(grid.nbytes)), end='\n')
properties.attrs['iterations'] += 1
# One last call to ensure 'properties' and 'grid' are consistent with one another
lower_convex_hull(grid, properties)
ravelled_X_view = grid['X'].values.view().reshape(-1, grid['X'].values.shape[-1])
ravelled_Y_view = grid['Y'].values.view().reshape(-1, grid['Y'].values.shape[-1])
ravelled_Phase_view = grid['Phase'].values.view().reshape(-1)
# Copy final point values from the grid and drop the index array
# For some reason direct construction doesn't work. We have to create empty and then assign.
properties['X'] = xray.DataArray(np.empty_like(ravelled_X_view[properties['points'].values]),
dims=properties['points'].dims + ('component',))
properties['X'].values[...] = ravelled_X_view[properties['points'].values]
properties['Y'] = xray.DataArray(np.empty_like(ravelled_Y_view[properties['points'].values]),
dims=properties['points'].dims + ('internal_dof',))
properties['Y'].values[...] = ravelled_Y_view[properties['points'].values]
# TODO: What about invariant reactions? We should perform a final driving force calculation here.
# We can handle that in the same post-processing step where we identify single-phase regions.
properties['Phase'] = xray.DataArray(np.empty_like(ravelled_Phase_view[properties['points'].values]),
dims=properties['points'].dims)
properties['Phase'].values[...] = ravelled_Phase_view[properties['points'].values]
del properties['points']
return properties
def equilibrium(dbf, comps, phases, conditions, **kwargs):
"""
Calculate the equilibrium state of a system containing the specified
components and phases, under the specified conditions.
Model parameters are taken from 'dbf'.
Parameters
----------
dbf : Database
Thermodynamic database containing the relevant parameters.
comps : list
Names of components to consider in the calculation.
phases : list or dict
Names of phases to consider in the calculation.
conditions : dict or (list of dict)
StateVariables and their corresponding value.
verbose : bool, optional (Default: True)
Show progress of calculations.
grid_opts : dict, optional
Keyword arguments to pass to the initial grid routine.
Returns
-------
Structured equilibrium calculation.
Examples
--------
None yet.
"""
active_phases = unpack_phases(phases) or sorted(dbf.phases.keys())
comps = sorted(comps)
indep_vars = ['T', 'P']
grid_opts = kwargs.pop('grid_opts', dict())
verbose = kwargs.pop('verbose', True)
phase_records = dict()
callable_dict = kwargs.pop('callables', dict())
grad_callable_dict = kwargs.pop('grad_callables', dict())
points_dict = dict()
maximum_internal_dof = 0
# Construct models for each phase; prioritize user models
models = unpack_kwarg(kwargs.pop('model', Model), default_arg=Model)
if verbose:
print('Components:', ' '.join(comps))
print('Phases:', end=' ')
for name in active_phases:
mod = models[name]
if isinstance(mod, type):
models[name] = mod = mod(dbf, comps, name)
variables = sorted(mod.energy.atoms(v.StateVariable).union({key for key in conditions.keys() if key in [v.T, v.P]}), key=str)
site_fracs = sorted(mod.energy.atoms(v.SiteFraction), key=str)
maximum_internal_dof = max(maximum_internal_dof, len(site_fracs))
# Extra factor '1e-100...' is to work around an annoying broadcasting bug for zero gradient entries
models[name].models['_broadcaster'] = 1e-100 * Mul(*variables) ** 3
out = models[name].energy
if name not in callable_dict:
undefs = list(out.atoms(Symbol) - out.atoms(v.StateVariable))
for undef in undefs:
out = out.xreplace({undef: float(0)})
# callable_dict takes variables in a different order due to calculate() pecularities
callable_dict[name] = make_callable(out,
sorted((key for key in conditions.keys() if key in [v.T, v.P]),
key=str) + site_fracs)
if name not in grad_callable_dict:
grad_func = make_callable(Matrix([out]).jacobian(variables), variables)
else:
grad_func = grad_callable_dict[name]
# Adjust gradient by the approximate chemical potentials
plane_vars = sorted(models[name].energy.atoms(v.SiteFraction), key=str)
hyperplane = Add(*[v.MU(i)*mole_fraction(dbf.phases[name], comps, i)
for i in comps if i != 'VA'])
# Workaround an annoying bug with zero gradient entries
# This forces numerically zero entries to broadcast correctly
hyperplane += 1e-100 * Mul(*([v.MU(i) for i in comps if i != 'VA'] + plane_vars + [v.T, v.P])) ** 3
plane_grad = make_callable(Matrix([hyperplane]).jacobian(variables),
[v.MU(i) for i in comps if i != 'VA'] + plane_vars + [v.T, v.P])
plane_hess = make_callable(hessian(hyperplane, variables),
[v.MU(i) for i in comps if i != 'VA'] + plane_vars + [v.T, v.P])
phase_records[name.upper()] = PhaseRecord(variables=variables,
grad=grad_func,
plane_grad=plane_grad,
plane_hess=plane_hess)
if verbose:
print(name, end=' ')
if verbose:
print('[done]', end='\n')
conds = OrderedDict((key, unpack_condition(value)) for key, value in sorted(conditions.items(), key=str))
str_conds = OrderedDict((str(key), value) for key, value in conds.items())
indep_vals = list([float(x) for x in np.atleast_1d(val)]
for key, val in str_conds.items() if key in indep_vars)
components = [x for x in sorted(comps) if not x.startswith('VA')]
# 'calculate' accepts conditions through its keyword arguments
grid_opts.update({key: value for key, value in str_conds.items() if key in indep_vars})
if 'pdens' not in grid_opts:
grid_opts['pdens'] = 10
coord_dict = str_conds.copy()
coord_dict['vertex'] = np.arange(len(components))
grid_shape = np.meshgrid(*coord_dict.values(),
indexing='ij', sparse=False)[0].shape
coord_dict['component'] = components
if verbose:
print('Computing initial grid', end=' ')
grid = calculate(dbf, comps, active_phases, output='GM',
model=models, callables=callable_dict, fake_points=True, **grid_opts)
if verbose:
print('[{0} points, {1}]'.format(len(grid.points), sizeof_fmt(grid.nbytes)), end='\n')
properties = xray.Dataset({'NP': (list(str_conds.keys()) + ['vertex'],
np.empty(grid_shape)),
'GM': (list(str_conds.keys()),
np.empty(grid_shape[:-1])),
'MU': (list(str_conds.keys()) + ['component'],
np.empty(grid_shape)),
'points': (list(str_conds.keys()) + ['vertex'],
np.empty(grid_shape, dtype=np.int))
},
coords=coord_dict,
attrs={'iterations': 1},
)
# Store the potentials from the previous iteration
current_potentials = properties.MU.copy()
for iteration in range(MAX_ITERATIONS):
if verbose:
print('Computing convex hull [iteration {}]'.format(properties.attrs['iterations']))
# lower_convex_hull will modify properties
lower_convex_hull(grid, properties)
progress = np.abs(current_potentials - properties.MU).max().values
if verbose:
print('progress', progress)
if progress < MIN_PROGRESS:
if verbose:
print('Convergence achieved')
break
current_potentials[...] = properties.MU.values
if verbose:
print('Refining convex hull')
# Insert extra dimensions for non-T,P conditions so GM broadcasts correctly
energy_broadcast_shape = grid.GM.values.shape[:len(indep_vals)] + \
(1,) * (len(str_conds) - len(indep_vals)) + (grid.GM.values.shape[-1],)
driving_forces = np.einsum('...i,...i',
properties.MU.values[..., np.newaxis, :],
grid.X.values[np.index_exp[...] +
(np.newaxis,) * (len(str_conds) - len(indep_vals)) +
np.index_exp[:, :]]) - \
grid.GM.values.view().reshape(energy_broadcast_shape)
for name in active_phases:
dof = len(models[name].energy.atoms(v.SiteFraction))
current_phase_indices = (grid.Phase.values == name).reshape(energy_broadcast_shape[:-1] + (-1,))
# Broadcast to capture all conditions
current_phase_indices = np.broadcast_arrays(current_phase_indices,
np.empty(driving_forces.shape))[0]
# This reshape is safe as long as phases have the same number of points at all indep. conditions
current_phase_driving_forces = driving_forces[current_phase_indices].reshape(
current_phase_indices.shape[:-1] + (-1,))
# Note: This works as long as all points are in the same phase order for all T, P
current_site_fractions = grid.Y.values[..., current_phase_indices[(0,) * len(str_conds)], :]
if np.sum(current_site_fractions[(0,) * len(indep_vals)][..., :dof]) == dof:
# All site fractions are 1, aka zero internal degrees of freedom
# Impossible to refine these points, so skip this phase
points_dict[name] = current_site_fractions[(0,) * len(indep_vals)][..., :dof]
continue
# Find the N points with largest driving force for a given set of conditions
# Remember that driving force has a sign, so we want the "most positive" values
# N is the number of components, in this context
# N points define a 'best simplex' for every set of conditions
# We also need to restrict ourselves to one phase at a time
trial_indices = np.argpartition(current_phase_driving_forces,
-len(components), axis=-1)[..., -len(components):]
trial_indices = trial_indices.ravel()
statevar_indices = np.unravel_index(np.arange(np.multiply.reduce(properties.GM.values.shape + (len(components),))),
properties.GM.values.shape + (len(components),))[:len(indep_vals)]
points = current_site_fractions[np.index_exp[statevar_indices + (trial_indices,)]]
points.shape = properties.points.shape[:-1] + (-1, maximum_internal_dof)
# The Y arrays have been padded, so we should slice off the padding
points = points[..., :dof]
# Workaround for derivative issues at endmembers
points[points == 0.] = MIN_SITE_FRACTION
if len(points) == 0:
if name in points_dict:
del points_dict[name]
# No nearly stable points: skip this phase
continue
num_vars = len(phase_records[name].variables)
plane_grad = phase_records[name].plane_grad
plane_hess = phase_records[name].plane_hess
statevar_grid = np.meshgrid(*itertools.chain(indep_vals), sparse=True, indexing='xy')
# TODO: A more sophisticated treatment of constraints
num_constraints = len(indep_vals) + len(dbf.phases[name].sublattices)
constraint_jac = np.zeros((num_constraints, num_vars))
# Independent variables are always fixed (in this limited implementation)
for idx in range(len(indep_vals)):
constraint_jac[idx, idx] = 1
# This is for site fraction balance constraints
var_idx = len(indep_vals)
for idx in range(len(dbf.phases[name].sublattices)):
active_in_subl = set(dbf.phases[name].constituents[idx]).intersection(comps)
constraint_jac[len(indep_vals) + idx,
var_idx:var_idx + len(active_in_subl)] = 1
var_idx += len(active_in_subl)
grad = phase_records[name].grad(*itertools.chain(statevar_grid, points.T))
if grad.dtype == 'object':
# Workaround a bug in zero gradient entries
grad_zeros = np.zeros(points.T.shape[1:], dtype=np.float)
for i in np.arange(grad.shape[0]):
if isinstance(grad[i], int):
grad[i] = grad_zeros
grad = np.array(grad.tolist(), dtype=np.float)
bcasts = np.broadcast_arrays(*itertools.chain(properties.MU.values.T, points.T))
cast_grad = -plane_grad(*itertools.chain(bcasts, [0], [0]))
cast_grad = cast_grad.T + grad.T
grad = cast_grad
grad.shape = grad.shape[:-1] # Remove extraneous dimension
# This Hessian is an approximation updated using the BFGS method
# See Nocedal and Wright, ch.3, p. 198
# Initialize as identity matrix
hess = broadcast_to(np.eye(num_vars), grad.shape + (grad.shape[-1],)).copy()
newton_iteration = 0
while newton_iteration < MAX_NEWTON_ITERATIONS:
e_matrix = np.linalg.inv(hess)
dy_unconstrained = -np.einsum('...ij,...j->...i', e_matrix, grad)
proj_matrix = np.dot(e_matrix, constraint_jac.T)
inv_matrix = np.rollaxis(np.dot(constraint_jac, proj_matrix), 0, -1)
inv_term = np.linalg.inv(inv_matrix)
first_term = np.einsum('...ij,...jk->...ik', proj_matrix, inv_term)
# Normally a term for the residual here
# We only choose starting points which obey the constraints, so r = 0
cons_summation = np.einsum('...i,...ji->...j', dy_unconstrained, constraint_jac)
cons_correction = np.einsum('...ij,...j->...i', first_term, cons_summation)
dy_constrained = dy_unconstrained - cons_correction
# TODO: Support for adaptive changing independent variable steps
new_direction = dy_constrained[..., len(indep_vals):]
# Backtracking line search
new_points = points + INITIAL_STEP_SIZE * new_direction
alpha = np.full(new_points.shape[:-1], INITIAL_STEP_SIZE, dtype=np.float)
negative_points = np.any(new_points < 0., axis=-1)
while np.any(negative_points):
alpha[negative_points] *= 0.1
new_points = points + alpha[..., np.newaxis] * new_direction
negative_points = np.any(new_points < 0., axis=-1)
# If we made "near" zero progress on any points, don't update the Hessian until
# we've rebuilt the convex hull
# Nocedal and Wright recommend against skipping Hessian updates
# They recommend using a damped update approach, pp. 538-539 of their book
# TODO: Check the projected gradient norm, not the step length
if np.any(np.max(np.abs(alpha[..., np.newaxis] * new_direction), axis=-1) < MIN_STEP_LENGTH):
break
# Workaround for derivative issues at endmembers
new_points[new_points == 0.] = 1e-16
# BFGS update to Hessian
new_grad = phase_records[name].grad(*itertools.chain(statevar_grid, new_points.T))
if new_grad.dtype == 'object':
# Workaround a bug in zero gradient entries
grad_zeros = np.zeros(new_points.T.shape[1:], dtype=np.float)
for i in np.arange(new_grad.shape[0]):
if isinstance(new_grad[i], int):
new_grad[i] = grad_zeros
new_grad = np.array(new_grad.tolist(), dtype=np.float)
bcasts = np.broadcast_arrays(*itertools.chain(properties.MU.values.T, new_points.T))
cast_grad = -plane_grad(*itertools.chain(bcasts, [0], [0]))
cast_grad = cast_grad.T + new_grad.T
new_grad = cast_grad
new_grad.shape = new_grad.shape[:-1] # Remove extraneous dimension
# Notation used here consistent with Nocedal and Wright
s_k = np.empty(points.shape[:-1] + (points.shape[-1] + len(indep_vals),))
# Zero out independent variable changes for now
s_k[..., :len(indep_vals)] = 0
s_k[..., len(indep_vals):] = new_points - points
y_k = new_grad - grad
s_s_term = np.einsum('...j,...k->...jk', s_k, s_k)
s_b_s_term = np.einsum('...i,...ij,...j', s_k, hess, s_k)
y_y_y_s_term = np.einsum('...j,...k->...jk', y_k, y_k) / \
np.einsum('...i,...i', y_k, s_k)[..., np.newaxis, np.newaxis]
update = np.einsum('...ij,...jk,...kl->...il', hess, s_s_term, hess) / \
s_b_s_term[..., np.newaxis, np.newaxis] + y_y_y_s_term
hess = hess - update
cast_hess = -plane_hess(*itertools.chain(bcasts, [0], [0])).T + hess
hess = -cast_hess #TODO: Why does this fix things?
# TODO: Verify that the chosen step lengths reduce the energy
points = new_points
grad = new_grad
newton_iteration += 1
new_points = new_points.reshape(new_points.shape[:len(indep_vals)] + (-1, new_points.shape[-1]))
new_points = np.concatenate((current_site_fractions[..., :dof], new_points), axis=-2)
points_dict[name] = new_points
if verbose:
print('Rebuilding grid', end=' ')
grid = calculate(dbf, comps, active_phases, output='GM',
model=models, callables=callable_dict,
fake_points=True, points=points_dict, **grid_opts)
if verbose:
print('[{0} points, {1}]'.format(len(grid.points), sizeof_fmt(grid.nbytes)), end='\n')
properties.attrs['iterations'] += 1
# One last call to ensure 'properties' and 'grid' are consistent with one another
lower_convex_hull(grid, properties)
ravelled_X_view = grid['X'].values.view().reshape(-1, grid['X'].values.shape[-1])
ravelled_Y_view = grid['Y'].values.view().reshape(-1, grid['Y'].values.shape[-1])
ravelled_Phase_view = grid['Phase'].values.view().reshape(-1)
# Copy final point values from the grid and drop the index array
# For some reason direct construction doesn't work. We have to create empty and then assign.
properties['X'] = xray.DataArray(np.empty_like(ravelled_X_view[properties['points'].values]),
dims=properties['points'].dims + ('component',))
properties['X'].values[...] = ravelled_X_view[properties['points'].values]
properties['Y'] = xray.DataArray(np.empty_like(ravelled_Y_view[properties['points'].values]),
dims=properties['points'].dims + ('internal_dof',))
properties['Y'].values[...] = ravelled_Y_view[properties['points'].values]
# TODO: What about invariant reactions? We should perform a final driving force calculation here.
# We can handle that in the same post-processing step where we identify single-phase regions.
properties['Phase'] = xray.DataArray(np.empty_like(ravelled_Phase_view[properties['points'].values]),
dims=properties['points'].dims)
properties['Phase'].values[...] = ravelled_Phase_view[properties['points'].values]
del properties['points']
return properties