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