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, **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())
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
