# Source code for burnman.tools.eos

```# This file is part of BurnMan - a thermoelastic and thermodynamic toolkit for
# the Earth and Planetary Sciences
# Copyright (C) 2012 - 2021 by the BurnMan team, released under the GNU
# GPL v2 or later.

import numpy as np
import warnings
from scipy.linalg import logm

[docs]def check_eos_consistency(m, P=1.e9, T=300., tol=1.e-4, verbose=False,
including_shear_properties=True):
"""
Compute numerical derivatives of the gibbs free energy of a mineral
under given conditions, and check these values against those provided
analytically by the equation of state

Parameters
----------
m : mineral
The mineral for which the equation of state
is to be checked for consistency
P : float
The pressure at which to check consistency
T : float
The temperature at which to check consistency
tol : float
The fractional tolerance for each of the checks
verbose : boolean
Decide whether to print information about each
check
including_shear_properties : boolean
Decide whether to check shear information,
which is pointless for liquids and equations of state
without shear modulus parameterizations

Returns
-------
consistency: boolean
If all checks pass, returns True

"""
dT = 1.
dP = 1000.

m.set_state(P, T)
G0 = m.gibbs
S0 = m.S
V0 = m.V

expr = ['G = F + PV', 'G = H - TS', 'G = E - TS + PV']
eq = [[m.gibbs, (m.helmholtz + P*m.V)],
[m.gibbs, (m.H - T*m.S)],
[m.gibbs, (m.molar_internal_energy - T*m.S + P*m.V)]]

m.set_state(P, T + dT)
G1 = m.gibbs
S1 = m.S
V1 = m.V

m.set_state(P + dP, T)
G2 = m.gibbs
V2 = m.V

# T derivatives
m.set_state(P, T + 0.5*dT)
expr.extend(['S = -dG/dT', 'alpha = 1/V dV/dT', 'C_p = T dS/dT'])
eq.extend([[m.S, -(G1 - G0)/dT],
[m.alpha, (V1 - V0)/dT/m.V],
[m.molar_heat_capacity_p, (T + 0.5*dT)*(S1 - S0)/dT]])

# P derivatives
m.set_state(P + 0.5*dP, T)
expr.extend(['V = dG/dP', 'K_T = -V dP/dV'])
eq.extend([[m.V, (G2 - G0)/dP],
[m.K_T, -0.5*(V2 + V0)*dP/(V2 - V0)]])

expr.extend(['C_v = Cp - alpha^2*K_T*V*T', 'K_S = K_T*Cp/Cv',
'gr = alpha*K_T*V/Cv'])
eq.extend([[m.molar_heat_capacity_v,
m.molar_heat_capacity_p - m.alpha*m.alpha*m.K_T*m.V*T],
[m.K_S, m.K_T*m.molar_heat_capacity_p/m.molar_heat_capacity_v],
[m.gr, m.alpha*m.K_T*m.V/m.molar_heat_capacity_v]])

expr.append('Vphi = np.sqrt(K_S/rho)')
eq.append([m.bulk_sound_velocity, np.sqrt(m.K_S/m.rho)])

if including_shear_properties:
expr.extend(['Vp = np.sqrt((K_S + 4G/3)/rho)',
'Vs = np.sqrt(G_S/rho)'])

with warnings.catch_warnings(record=True) as w:
warnings.simplefilter("always")
eq.extend([[m.p_wave_velocity, np.sqrt((m.K_S + 4.*m.G/3.)/m.rho)],
[m.shear_wave_velocity, np.sqrt(m.G/m.rho)]])
if len(w) == 1:
print(w[0].message)
print('\nYou can suppress this message by setting the '
'parameter\nincluding_shear_properties to False '
'when calling check_eos_consistency.\n')
note = ''
else:
note = ' (not including shear properties)'

consistencies = [np.abs(e[0] - e[1])
< np.abs(tol*e[1]) + np.finfo('float').eps for e in eq]
eos_is_consistent = np.all(consistencies)

if verbose:
print('Checking EoS consistency for {0:s}{1}'.format(m.to_string(),
note))
print('Expressions within tolerance of {0:2f}'.format(tol))
for i, c in enumerate(consistencies):
print('{0:10s} : {1:5s}'.format(expr[i], str(c)))
if eos_is_consistent:
print('All EoS consistency constraints satisfied for {0:s}'.format(m.to_string()))
else:
print('Not satisfied all EoS consistency constraints for {0:s}'.format(m.to_string()))

return eos_is_consistent

[docs]def check_anisotropic_eos_consistency(m, P=1.e9, T=2000.,
tol=1.e-4, verbose=False):
"""
Compute numerical derivatives of the gibbs free energy of a mineral
under given conditions, and check these values against those provided
analytically by the equation of state

Parameters
----------
m : mineral
The mineral for which the equation of state
is to be checked for consistency
P : float
The pressure at which to check consistency
T : float
The temperature at which to check consistency
tol : float
The fractional tolerance for each of the checks
verbose : boolean
Decide whether to print information about each
check

Returns
-------
consistency: boolean
If all checks pass, returns True

"""
dT = 1.
dP = 1000.

m.set_state(P, T)
G0 = m.gibbs
S0 = m.S
V0 = m.V

expr = ['G = F + PV', 'G = H - TS', 'G = E - TS + PV']
eq = [[m.gibbs, (m.helmholtz + P*m.V)],
[m.gibbs, (m.H - T*m.S)],
[m.gibbs, (m.molar_internal_energy - T*m.S + P*m.V)]]

m.set_state(P, T + dT)
G1 = m.gibbs
S1 = m.S
V1 = m.V

m.set_state(P + dP, T)
G2 = m.gibbs
V2 = m.V

# T derivatives
m.set_state(P, T + 0.5*dT)
expr.extend(['S = -dG/dT', 'alpha = 1/V dV/dT', 'C_p = T dS/dT'])
eq.extend([[m.S, -(G1 - G0)/dT],
[m.alpha, (V1 - V0)/dT/m.V],
[m.molar_heat_capacity_p, (T + 0.5*dT)*(S1 - S0)/dT]])

# P derivatives
m.set_state(P + 0.5*dP, T)
expr.extend(['V = dG/dP', 'K_T = -V dP/dV'])
eq.extend([[m.V, (G2 - G0)/dP],
[m.isothermal_bulk_modulus_reuss, -0.5*(V2 + V0)*dP/(V2 - V0)]])

expr.extend(['C_v = Cp - alpha^2*K_T*V*T', 'K_S = K_T*Cp/Cv'])
eq.extend([[m.molar_heat_capacity_v,
m.molar_heat_capacity_p - m.alpha*m.alpha*m.K_T*m.V*T],
[m.isentropic_bulk_modulus_reuss,
m.isothermal_bulk_modulus_reuss
* m.molar_heat_capacity_p / m.molar_heat_capacity_v]])

# Third derivative
m.set_state(P + 0.5 * dP, T)
b0 = m.isothermal_compressibility_tensor

m.set_state(P + 0.5 * dP, T + dT)
b1 = m.isothermal_compressibility_tensor

m.set_state(P, T + 0.5 * dT)
a0 = m.thermal_expansivity_tensor

m.set_state(P + dP, T + 0.5 * dT)
a1 = m.thermal_expansivity_tensor

m.set_state(P + 0.5 * dP, T + 0.5 * dT)

beta0 = -(logm(F3) - logm(F2))/dP
alpha0 = (logm(F1) - logm(F0))/dT

Q = m.deformed_coordinate_frame
beta1 = m.isothermal_compressibility_tensor
alpha1 = m.thermal_expansivity_tensor

beta1 = np.einsum('mi, nj, ij->mn', Q, Q, beta1)
alpha1 = np.einsum('mi, nj, ij->mn', Q, Q, alpha1)

expr.extend([f'SI = -d(lnm(F))/dP ({i}{j})'
for i in range(3) for j in range(i, 3)])
eq.extend([[beta0[i, j], beta1[i, j]]
for i in range(3) for j in range(i, 3)])

expr.extend([f'alpha = d(lnm(F))/dT ({i}{j})'
for i in range(3) for j in range(i, 3)])
eq.extend([[alpha0[i, j], alpha1[i, j]]
for i in range(3) for j in range(i, 3)])

expr.extend([f'd(alpha)/dP = -d(beta_T)/dT ({i}{j})'
for i in range(3) for j in range(i, 3)])
eq.extend([[(a1[i, j] - a0[i, j])/dP, -(b1[i, j] - b0[i, j])/dT]
for i in range(3) for j in range(i, 3)])

# Consistent Phi
expr.extend(['dPsidf_Voigt[:3,:3] == 1'])
eq.extend([[np.sum(m.dPsidf_Voigt[:3,:3]), 1.]])

# Consistent inverses
expr.extend([f'S_T = inv(C_T) ({i}{j})'
for i in range(6) for j in range(i, 6)])
S_T = m.isothermal_compliance_tensor
S_T2 = np.linalg.inv(m.isothermal_stiffness_tensor)
eq.extend([[S_T[i,j], S_T2[i,j]]
for i in range(6) for j in range(i, 6)])

expr.extend([f'S_N = inv(C_N) ({i}{j})'
for i in range(6) for j in range(i, 6)])
S_N = m.isentropic_compliance_tensor
S_N2 = np.linalg.inv(m.isentropic_stiffness_tensor)
eq.extend([[S_N[i,j], S_N2[i,j]]
for i in range(6) for j in range(i, 6)])

# Consistent isotropic and anisotropic properties
expr.extend(['V = det(M)',
'alpha_v = tr(alpha)',
'beta_T = sum(S_T I)',
'beta_S = sum(S_S I)'])
eq.extend([[m.V, np.linalg.det(m.cell_vectors)],
[m.alpha, np.trace(m.thermal_expansivity_tensor)],
[m.beta_T, np.sum(m.isothermal_compliance_tensor[:3, :3])],
[m.beta_S, np.sum(m.isentropic_compliance_tensor[:3, :3])]])

expr.append('Vphi = np.sqrt(K_S/rho)')
eq.append([m.bulk_sound_velocity, np.sqrt(m.K_S/m.rho)])

consistencies = [np.abs(e[0] - e[1]) < np.abs(tol*e[1])
+ np.finfo('float').eps for e in eq]
eos_is_consistent = np.all(consistencies)

if verbose:
print('Checking EoS consistency for {0:s}'.format(m.to_string()))
print('Expressions within tolerance of {0:2f}'.format(tol))
for i, c in enumerate(consistencies):
print('{0:10s} : {1:5s}'.format(expr[i], str(c)))
if eos_is_consistent:
print('All EoS consistency constraints satisfied for {0:s}'.format(m.to_string()))
else:
print('Not satisfied all EoS consistency constraints for {0:s}'.format(m.to_string()))

return eos_is_consistent
```