```"""
The calculate module contains a routine for calculating the
property surface of a system.
"""

import itertools
import warnings
from collections import OrderedDict
from collections.abc import Mapping
import numpy as np
from pycalphad.core.utils import endmember_matrix, extract_parameters, \
get_pure_elements, filter_phases, instantiate_models, point_sample, \
unpack_components, unpack_condition, unpack_kwarg

[docs]def hr_point_sample(constraint_jac, constraint_rhs, initial_point, num_points):
"Hit-and-run sampling of linearly-constrained site fraction spaces"
q, r = np.linalg.qr(constraint_jac.T, mode='complete')
q1 = q[:, :constraint_jac.shape]
q2 = q[:, constraint_jac.shape:]
r1 = r[:constraint_jac.shape, :]
if initial_point is not None:
z_bar = initial_point
else:
# minimum norm solution to underdetermined system of equations
# may not be feasible if it fails the non-negativity constraint
z_bar = np.linalg.lstsq(constraint_jac, constraint_rhs, rcond=None)
solution_norm = np.linalg.norm(constraint_jac.dot(z_bar) - constraint_rhs)
if (solution_norm > 1e-4) or np.any(z_bar < 0):
# initial point does not satisfy constraints; give up
return np.empty((0, z_bar.shape))
# Hit-and-Run sampling
new_feasible_z = np.zeros((num_points, constraint_jac.shape))
current_z = np.array(z_bar)
min_z = MIN_SITE_FRACTION
rng = np.random.RandomState(1769)
for iteration in range(num_points):
# generate unit direction in null space
d = rng.normal(size=(constraint_jac.shape - constraint_jac.shape))
d /= np.linalg.norm(d, axis=0)
proj = np.dot(q2, d)
# find extent of step direction possible while staying within bounds (0 <= z)
with np.errstate(divide='ignore'):
alphas = (min_z - current_z) / proj
# Need to use small value to prevent constraints binding one sublattice (with proj ~ 0) from binding all dof
max_alpha_candidates = alphas[np.logical_and(proj > 1e-6, np.isfinite(alphas))]
min_alpha_candidates = alphas[np.logical_and(proj < -1e-6, np.isfinite(alphas))]
alpha_min = np.min(min_alpha_candidates)
alpha_max = np.max(max_alpha_candidates)
# Poor progress; give up on sampling
if np.abs(alpha_max - alpha_min) < 1e-4:
new_feasible_z = new_feasible_z[:iteration, :]
break
# choose a random step size within the feasible interval
new_alpha = rng.uniform(low=alpha_min, high=alpha_max)
current_z += new_alpha * proj
new_feasible_z[iteration, :] = current_z
if np.any(new_feasible_z < 0):
raise ValueError('Constrained sampling generated negative site fractions')
return new_feasible_z

@cacheit
def _sample_phase_constitution(model, sampler, fixed_grid, pdens):
"""
Sample the internal degrees of freedom of a phase.

Parameters
----------
model : Model
sampler : Callable
Callable returning an ArrayLike of points
fixed_grid : bool
If True, sample pdens points between each pair of endmembers
pdens : int
Number of points to sample in each sampled dimension

Returns
-------
ndarray of points
"""
# Eliminate pure vacancy endmembers from the calculation
ALLOWED_CHARGE=1E-10
vacancy_indices = []
for sublattice in model.constituents:
subl_va_indices = [idx for idx, spec in enumerate(sorted(set(sublattice))) if spec.number_of_atoms == 0]
vacancy_indices.append(subl_va_indices)
if len(vacancy_indices) != len(model.constituents):
vacancy_indices = None
sublattice_dof = [len(subl) for subl in model.constituents]
# Add all endmembers to guarantee their presence
points = endmember_matrix(sublattice_dof, vacancy_indices=vacancy_indices)
site_ratios = model.site_ratios
constant_site_ratios = True
# The only implementation with variable site ratios is the two-sublattice ionic liquid.
# This check is convenient for detecting 2SL ionic liquids without keeping other state.
for sr in site_ratios:
try:
float(sr)
except (TypeError, RuntimeError):
constant_site_ratios = False
species_charge = []
for sublattice in range(len(model.constituents)):
for species in sorted(model.constituents[sublattice]):
species_charge.append(species.charge*site_ratios[sublattice])
species_charge = np.array(species_charge)
charge_constrained_space = constant_site_ratios and np.any(species_charge != 0)
# We differentiate between (specifically) charge balance and general linear constraints for future use
# This simplifies adding future constraints, such as disordered configuration sampling, or site fraction conditions
# Note that if a phase only consists of site fraction balance constraints,
# we do not consider that 'linearly constrained' for the purposes of sampling,
# since the default sampler handles that case with an efficient method.
linearly_constrained_space = charge_constrained_space

if charge_constrained_space:
endmembers = points
Q = np.dot(endmembers, species_charge)
# Sort endmembers by their charge
charge_neutral_endmember_idxs = []
charge_positive_endmember_idxs = []
charge_negative_endmember_idxs = []
for em_idx in range(endmembers.shape):
if Q[em_idx] > ALLOWED_CHARGE:
charge_positive_endmember_idxs.append(em_idx)
elif Q[em_idx] < -ALLOWED_CHARGE:
charge_negative_endmember_idxs.append(em_idx)
else:
charge_neutral_endmember_idxs.append(em_idx)

# Find all endmember pairs between the
em_pts = [endmembers[em_idx] for em_idx in charge_neutral_endmember_idxs]
for pos_em_idx, neg_em_idx in itertools.product(charge_positive_endmember_idxs, charge_negative_endmember_idxs):
# Solve equation: Q_{pos}*x + Q_{neg}(1-x) = 0
x = - Q[neg_em_idx] / (Q[pos_em_idx] - Q[neg_em_idx])
em_pts.append(endmembers[pos_em_idx] * x + endmembers[neg_em_idx] * (1-x))

# Charge neutral endmembers and mixed pseudo-endmembers
points = np.asarray(em_pts)

if (fixed_grid is True) and not linearly_constrained_space:
# Sample along the edges of the endmembers
# These constitution space edges are often the equilibrium points!
em_pairs = list(itertools.combinations(points, 2))
lingrid = np.linspace(0, 1, pdens)
extra_points = [first_em * lingrid[np.newaxis].T +
second_em * lingrid[::-1][np.newaxis].T
for first_em, second_em in em_pairs]
points = np.concatenate(list(itertools.chain([points], extra_points)))
# Sample composition space for more points
if sum(sublattice_dof) > len(sublattice_dof):
if linearly_constrained_space:
# construct constraint Jacobian for this phase
# Model technically already does this so it would be better to reuse that functionality
# number of sublattices, plus charge balance
num_constraints = len(sublattice_dof) + 1
constraint_jac = np.zeros((num_constraints, points.shape[-1]))
constraint_rhs = np.zeros(num_constraints)
# site fraction balance
dof_idx = 0
constraint_idx = 0
for subl_dof in sublattice_dof:
constraint_jac[constraint_idx, dof_idx:dof_idx + subl_dof] = 1
constraint_rhs[constraint_idx] = 1
constraint_idx += 1
dof_idx += subl_dof
# charge balance
constraint_jac[constraint_idx, :] = species_charge
constraint_rhs[constraint_idx] = 0
# Sample additional points which obey the constraints
# Mean of pseudo-endmembers is feasible by convexity of the space
initial_point = np.mean(points, axis=0)
num_points = (pdens ** 2) * (constraint_jac.shape - constraint_jac.shape)
extra_points = hr_point_sample(constraint_jac, constraint_rhs, initial_point, num_points)
points = np.concatenate((points, extra_points))
assert np.max(np.abs(constraint_jac.dot(points.T).T - constraint_rhs)) < 1e-6
if points.shape == 0:
warnings.warn(f'{model.phase_name} has zero feasible configurations under the given conditions')
else:
points = np.concatenate((points, sampler(sublattice_dof, pdof=pdens)))

# 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_))
return points

def _compute_phase_values(components, statevar_dict,
parameters=None, fake_points=False,
largest_energy=None):
"""
Calculate output values for a particular phase.

Parameters
----------
components : list
Names of components to consider in the calculation.
statevar_dict : OrderedDict {str -> float or sequence}
Mapping of state variables to desired values. This will broadcast if necessary.
points : ndarray
Inputs to 'func', except state variables. Columns should be in 'variables' order.
phase_record : PhaseRecord
Contains callable for energy and phase metadata.
output : string
Desired name of the output result in the Dataset.
maximum_internal_dof : int
Largest number of internal degrees of freedom of any phase. This is used
to guarantee different phase's Datasets can be concatenated.
If True, broadcast state variables against each other to create a grid.
If False, assume state variables are given as equal-length lists (or single-valued).
parameters : OrderedDict {str -> float or sequence}, optional
Maps SymEngine symbols to a scalar or 1-D array. The arrays must be equal length.
The corresponding PhaseRecord must have been initialized with the same parameters.
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.

Returns
-------
Dataset of the output attribute as a function of state variables

Examples
--------
None yet.
"""
# Broadcast compositions and state variables along orthogonal axes
# This lets us eliminate an expensive Python loop
statevars = np.meshgrid(*itertools.chain(statevar_dict.values(),
[np.empty(points.shape[-2])]),
sparse=True, indexing='ij')[:-1]
points = broadcast_to(points, tuple(len(np.atleast_1d(x)) for x in statevar_dict.values()) + points.shape[-2:])
else:
statevars = list(np.atleast_1d(x) for x in statevar_dict.values())
statevars_ = []
for statevar in statevars:
if (len(statevar) != len(points)) and (len(statevar) == 1):
statevar = np.repeat(statevar, len(points))
if (len(statevar) != len(points)) and (len(statevar) != 1):
raise ValueError('Length of state variable list and number of given points must be equal when '
statevars_.append(statevar)
statevars = statevars_
pure_elements = [list(x.constituents.keys()) for x in components]
pure_elements = sorted(set([el.upper() for constituents in pure_elements for el in constituents]))
pure_elements = [x for x in pure_elements if x != 'VA']
# func may only have support for vectorization along a single axis (no broadcasting)
# we need to force broadcasting and flatten the result before calling
bc_statevars = np.ascontiguousarray([broadcast_to(x, points.shape[:-1]).reshape(-1) for x in statevars])
pts = points.reshape(-1, points.shape[-1])
dof = np.ascontiguousarray(np.concatenate((bc_statevars.T, pts), axis=1))
phase_compositions = np.zeros((dof.shape, len(pure_elements)), order='F')

param_symbols, parameter_array = extract_parameters(parameters)
parameter_array_length = parameter_array.shape
if parameter_array_length == 0:
# No parameters specified
phase_output = np.zeros(dof.shape, order='C')
phase_record.obj_2d(phase_output, dof)
else:
# Vectorized parameter arrays
phase_output = np.zeros((dof.shape, parameter_array_length), order='C')
phase_record.obj_parameters_2d(phase_output, dof, parameter_array)

for el_idx in range(len(pure_elements)):
phase_record.mass_obj_2d(phase_compositions[:, el_idx], dof, el_idx)

max_tieline_vertices = len(pure_elements)
if isinstance(phase_output, (float, int)):
if isinstance(phase_compositions, (float, int)):
phase_compositions = broadcast_to(phase_output, points.shape[:-1] + (len(pure_elements),))
phase_output = np.asarray(phase_output, dtype=np.float_)
if parameter_array_length <= 1:
phase_output.shape = points.shape[:-1]
else:
phase_output.shape = points.shape[:-1] + (parameter_array_length,)
phase_compositions = np.asarray(phase_compositions, dtype=np.float_)
phase_compositions.shape = points.shape[:-1] + (len(pure_elements),)
if fake_points:
output_shape = points.shape[:-2] + (max_tieline_vertices,)
if parameter_array_length > 1:
output_shape = output_shape + (parameter_array_length,)
concat_axis = -2
else:
concat_axis = -1
phase_output = np.concatenate((broadcast_to(largest_energy, output_shape), phase_output), axis=concat_axis)
phase_names = np.concatenate((broadcast_to('_FAKE_', points.shape[:-2] + (max_tieline_vertices,)),
np.full(points.shape[:-1], phase_record.phase_name, dtype='U' + str(len(phase_record.phase_name)))), axis=-1)
else:
phase_names = np.full(points.shape[:-1], phase_record.phase_name, dtype='U'+str(len(phase_record.phase_name)))
if fake_points:
phase_compositions = np.concatenate((np.broadcast_to(np.eye(len(pure_elements)), points.shape[:-2] + (max_tieline_vertices, len(pure_elements))), phase_compositions), axis=-2)

coordinate_dict = {'component': pure_elements}
# Resize 'points' so it has the same number of columns as the maximum
# number of internal degrees of freedom of any phase in the calculation.
# We do this so that everything is aligned for concat.
# Waste of memory? Yes, but the alternatives are unclear.
# In each case, first check if we need to do this...
# It can be expensive for many points (~14s for 500M points)
if fake_points:
desired_shape = points.shape[:-2] + (max_tieline_vertices + points.shape[-2], maximum_internal_dof)
expanded_points = np.full(desired_shape, np.nan)
expanded_points[..., len(pure_elements):, :points.shape[-1]] = points
else:
desired_shape = points.shape[:-1] + (maximum_internal_dof,)
if points.shape == desired_shape:
expanded_points = points
else:
# TODO: most optimal solution would be to take pre-extended arrays as an argument and remove this
# This still copies the array, but is more efficient than filling
# an array with np.nan, then copying the existing points
append_nans = np.full(desired_shape[:-1] + (desired_shape[-1] - points.shape[-1],), np.nan)
expanded_points = np.append(points, append_nans, axis=-1)
coordinate_dict.update({key: np.atleast_1d(value) for key, value in statevar_dict.items()})
output_columns = [str(x) for x in statevar_dict.keys()] + ['points']
else:
output_columns = ['points']
if parameter_array_length > 1:
parameter_column = ['samples']
coordinate_dict['param_symbols'] = [str(x) for x in param_symbols]
else:
parameter_column = []
data_arrays = {'X': (output_columns + ['component'], np.ascontiguousarray(phase_compositions)),
'Phase': (output_columns, phase_names),
'Y': (output_columns + ['internal_dof'], expanded_points),
output: (['dim_'+str(i) for i in range(len(phase_output.shape) - (len(output_columns)+len(parameter_column)))] + output_columns + parameter_column, phase_output)
}
# Add state variables as data variables rather than as coordinates
for sym, vals in zip(statevar_dict.keys(), statevars):
data_arrays.update({sym: (output_columns, vals)})
if parameter_array_length > 1:
data_arrays['param_values'] = (['samples', 'param_symbols'], parameter_array)
return LightDataset(data_arrays, coords=coordinate_dict)

[docs]def calculate(dbf, comps, phases, mode=None, output='GM', fake_points=False, broadcast=True, parameters=None, to_xarray=True, phase_records=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.
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 SymEngine Symbol to numbers, for overriding the values of parameters in the Database.
phase_records : Optional[Mapping[str, PhaseRecord]]
Mapping of phase names to PhaseRecord objects. Must include all active phases.
The `model` argument must be a mapping of phase names to instances of Model
objects. Callers must take care that the PhaseRecord objects were created with
the same `output` as passed to `calculate`.

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)
callables = kwargs.pop('callables', {})
sampler_dict = unpack_kwarg(kwargs.pop('sampler', None), default_arg=None)
fixedgrid_dict = unpack_kwarg(kwargs.pop('grid_points', True), default_arg=True)
model = kwargs.pop('model', None)
parameters = parameters or dict()
if isinstance(parameters, dict):
parameters = OrderedDict(sorted(parameters.items(), key=str))
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]
nonvacant_elements = get_pure_elements(dbf, comps)

all_phase_data = []
largest_energy = 1e10

# Consider only the active phases
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 = 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(output, (list, tuple, set)):
raise NotImplementedError('Only one property can be specified in calculate() at a time')
output = output if output is not None else 'GM'

# Implicitly add 'N' state variable as a string to keyword arguements if it's not passed
if kwargs.get('N') is None:
kwargs['N'] = 1
if np.any(np.array(kwargs['N']) != 1):
raise ConditionError('N!=1 is not yet supported, got N={}'.format(kwargs['N']))

# TODO: conditions dict of StateVariable instances should become part of the calculate API
statevar_strings = [sv for sv in kwargs.keys() if getattr(v, sv) is not None]
# 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() if key in statevar_strings)
# Sort after default state variable check to fix gh-116
statevar_dict = OrderedDict(sorted(statevar_dict.items(), key=lambda x: str(x)))
str_statevar_dict = OrderedDict((str(key), unpack_condition(value)) for (key, value) in statevar_dict.items())

# Build phase records if they weren't passed
if phase_records is None:
models = instantiate_models(dbf, comps, active_phases, model=model, parameters=parameters)
phase_records = build_phase_records(dbf, comps, active_phases, statevar_dict,
models=models, parameters=parameters,
output=output, callables=callables,
verbose=kwargs.pop('verbose', False))
else:
# phase_records were provided, instantiated models must also be provided by the caller
models = model
if not isinstance(models, Mapping):
raise ValueError("A dictionary of instantiated models must be passed to `equilibrium` with the `model` argument if the `phase_records` argument is used.")
active_phases_without_models = [name for name in active_phases if not isinstance(models.get(name), Model)]
active_phases_without_phase_records = [name for name in active_phases if not isinstance(phase_records.get(name), PhaseRecord)]
if len(active_phases_without_phase_records) > 0:
raise ValueError(f"phase_records must contain a PhaseRecord instance for every active phase. Missing PhaseRecord objects for {sorted(active_phases_without_phase_records)}")
if len(active_phases_without_models) > 0:
raise ValueError(f"model must contain a Model instance for every active phase. Missing Model objects for {sorted(active_phases_without_models)}")

maximum_internal_dof = max(len(models[phase_name].site_fractions) for phase_name in active_phases)
for phase_name in sorted(active_phases):
mod = models[phase_name]
phase_record = phase_records[phase_name]
points = points_dict[phase_name]
if points is None:
points = _sample_phase_constitution(mod, 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))
phase_ds = _compute_phase_values(nonvacant_components, str_statevar_dict,
points, phase_record, output,
largest_energy=float(largest_energy), fake_points=fp)
all_phase_data.append(phase_ds)

fp_offset = len(nonvacant_elements) if fake_points else 0
running_total = [fp_offset] + list(np.cumsum([phase_ds['X'].shape[-2] for phase_ds in all_phase_data]))
islice_by_phase = {phase_name: slice(running_total[phase_idx], running_total[phase_idx+1], None)
for phase_idx, phase_name in enumerate(sorted(active_phases))}
# speedup for single-phase case (found by profiling)
if len(all_phase_data) > 1:
concatenated_coords = all_phase_data.coords

data_vars = all_phase_data.data_vars
concatenated_data_vars = {}
for var in data_vars.keys():
data_coords = data_vars[var]
points_idx = data_coords.index('points')  # concatenation axis
arrs = []
for phase_data in all_phase_data:
arrs.append(getattr(phase_data, var))
concat_data = np.concatenate(arrs, axis=points_idx)
concatenated_data_vars[var] = (data_coords, concat_data)
final_ds = LightDataset(data_vars=concatenated_data_vars, coords=concatenated_coords)
else:
final_ds = all_phase_data
final_ds.attrs['phase_indices'] = islice_by_phase
if to_xarray:
return final_ds.get_dataset()
else:
return final_ds
```