Source code for pycalphad.core.calculate

"""
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 numpy import broadcast_to
import pycalphad.variables as v
from pycalphad import ConditionError
from pycalphad.codegen.callables import build_phase_records
from pycalphad.core.cache import cacheit
from pycalphad.core.light_dataset import LightDataset
from pycalphad.model import Model
from pycalphad.core.phase_rec import PhaseRecord
from pycalphad.core.utils import endmember_matrix, extract_parameters, \
    get_pure_elements, filter_phases, instantiate_models, point_sample, \
    unpack_components, unpack_condition, unpack_kwarg
from pycalphad.core.constants import MIN_SITE_FRACTION


[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[0]] q2 = q[:, constraint_jac.shape[0]:] r1 = r[:constraint_jac.shape[0], :] 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)[0] 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[0])) # Hit-and-Run sampling new_feasible_z = np.zeros((num_points, constraint_jac.shape[1])) 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[1] - constraint_jac.shape[0])) 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 Instance of a pycalphad 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[0]): 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[1] - constraint_jac.shape[0]) 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] == 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, points, phase_record, output, maximum_internal_dof, broadcast=True, 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. broadcast : bool 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. """ if broadcast: # 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 ' 'broadcast=False.') 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[0], len(pure_elements)), order='F') param_symbols, parameter_array = extract_parameters(parameters) parameter_array_length = parameter_array.shape[0] if parameter_array_length == 0: # No parameters specified phase_output = np.zeros(dof.shape[0], order='C') phase_record.obj_2d(phase_output, dof) else: # Vectorized parameter arrays phase_output = np.zeros((dof.shape[0], 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)): phase_output = broadcast_to(phase_output, points.shape[:-1]) 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) if broadcast: 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) } if not broadcast: # 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. 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 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[0]))) 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, build_gradients=False, build_hessians=False, 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)[0]) phase_ds = _compute_phase_values(nonvacant_components, str_statevar_dict, points, phase_record, output, maximum_internal_dof, broadcast=broadcast, parameters=parameters, 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[0].coords data_vars = all_phase_data[0].data_vars concatenated_data_vars = {} for var in data_vars.keys(): data_coords = data_vars[var][0] 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[0] final_ds.attrs['phase_indices'] = islice_by_phase if to_xarray: return final_ds.get_dataset() else: return final_ds