def test_surface_tension_derivation():
"""Computes the excess free energy per unit area of an interface transition between two phases
which should give exactly the surface tension parameter"""
num_phases = 4
eta = sp.Symbol("eta")
free_energy = free_energy_functional_n_phases(num_phases,
interface_width=eta)
phi = symbolic_order_parameters(num_phases)
x = sp.Symbol("x")
sol = analytic_interface_profile(x, interface_width=eta)
for a, b in [(1, 3), (0, 1)]:
substitutions = {phi[a]: sol}
if b < len(phi) - 1:
substitutions[phi[b]] = 1 - sol
for i, phi_i in enumerate(phi[:-1]):
if i not in (a, b):
substitutions[phi_i] = 0
free_energy_2_phase = sp.simplify(
evaluate_diffs(free_energy.subs(substitutions), x))
result = cosh_integral(free_energy_2_phase, x)
assert result == symmetric_symbolic_surface_tension(a, b)
def create_n_phase_model_penalty_term(alpha=1,
num_phases=4,
kappa=0.015,
penalty_term_factor=0.01,
**kwargs):
order_parameters = symbolic_order_parameters(num_phases)
free_energy = free_energy_functional_n_phases_penalty_term(
order_parameters, alpha, kappa, penalty_term_factor)
return PhaseFieldStep(free_energy, order_parameters, **kwargs)
def test_analytic_interface_solution():
"""Ensures that the tanh is an analytical solution for the prescribed free energy / chemical potential
"""
num_phases = 4
phi = symbolic_order_parameters(num_phases - 1)
free_energy = free_energy_functional_n_phases(num_phases,
order_parameters=phi).subs(
{p: 0
for p in phi[1:]})
mu_diff_eq = chemical_potentials_from_free_energy(free_energy, [phi[0]])[0]
x = sp.Symbol("x")
sol = analytic_interface_profile(x)
inserted = mu_diff_eq.subs(phi[0], sol)
assert sp.expand(evaluate_diffs(inserted, x)) == 0
def create_n_phase_model(alpha=1,
num_phases=4,
surface_tensions=lambda i, j: 0.005 if i != j else 0,
f1=lambda c: c**2 * (1 - c)**2,
f2=lambda c: c**2 * (1 - c)**2,
triple_point_energy=0,
**kwargs):
order_parameters = symbolic_order_parameters(num_phases - 1)
free_energy = free_energy_functional_n_phases(
num_phases,
surface_tensions,
alpha,
order_parameters,
f1=f1,
f2=f2,
triple_point_energy=triple_point_energy)
def concentration_to_order_parameters(c):
c = np.array(c)
c_sum = np.sum(c, axis=-1)
if isinstance(c_sum, np.ndarray):
np.subtract(c_sum, 1, out=c_sum)
np.abs(c_sum, out=c_sum)
deviation = np.max(c_sum)
else:
deviation = np.abs(c_sum - 1)
if deviation > 1e-5:
warnings.warn(
"Initialization problem: concentrations have to add up to 1")
return c[..., :-1]
def order_parameters_to_concentrations(op):
op_shape = list(op.shape)
op_shape[-1] += 1
result = np.empty(op_shape)
np.copyto(result[..., :-1], op)
result[..., -1] = 1 - np.sum(op, axis=-1)
return result
return PhaseFieldStep(
free_energy,
order_parameters,
concentration_to_order_parameters=concentration_to_order_parameters,
order_parameters_to_concentrations=order_parameters_to_concentrations,
**kwargs)
def test_pressure_tensor():
"""
Checks that the following ways are equivalent:
1) phi -> mu -> force
2) phi -> pressure tensor -> force
"""
dim = 3
c = symbolic_order_parameters(3)
f = free_energy_functional_n_phases(order_parameters=c)
mu = chemical_potentials_from_free_energy(f, c)
mu = substitute_laplacian_by_sum(mu, dim)
force_chem_pot = expand_diff_full(force_from_phi_and_mu(c, dim, mu),
functions=c)
p = pressure_tensor_from_free_energy(f, c, dim)
force_pressure_tensor = force_from_pressure_tensor(p, functions=c)
for f1_i, f2_i in zip(force_chem_pot, force_pressure_tensor):
assert sp.expand(f1_i - f2_i) == 0