Effective diffusivity regime#
This verification case consists of a slab of depth \(l = 1 \times 10^{-3} \ \mathrm{m}\) with one trap under the effective diffusivity regime.
A trapping parameter \(\zeta\) is defined by
\[
\zeta = \frac{\lambda ^ 2 \nu}{D_0 \rho} \exp \left(\frac{E_k - E_p}{k_B T}\right) + \frac{c_m}{\rho}
\]
where
\(\lambda \ \mathrm{(m)}\) is the lattice parameter,
\(\nu \ (\mathrm{s}^{-1})\), the Debye frequency
\(\rho\) is the trapping site fraction,
\(c_m (\text{atom} \ \mathrm{m}^{-3})\) is the mobile atom concentration,
and the effective diffusivity \(\mathrm{D_\text{eff}}\) is defined by
\[
D_\text{eff} = \frac{D}{1 + \frac{1}{\zeta}}
\]
Then with a breakthrough time \(\tau = \frac{l^2}{2\pi^2 D_\text{eff}}\), the exact solution for flux is
\[
J = \frac{}{} \left[ 1 + 2\sum_{m=1}^\infty (-1)^m \exp \left( -m^2 \frac{t}{\tau} \right) \right]
\]
This analytical solution was obtained from TMAP7’s V&V report [3], case Val-1da.
FESTIM Code#
Show code cell content
# referenced https://github.com/gabriele-ferrero/Titans_TT_codecomparison/blob/main/Festim_models/WeakTrap.py
import festim as F
import numpy as np
import matplotlib.pyplot as plt
D_0 = 1.9e-7
N_A = 6.0221408e23
rho_w = 6.3382e28
E_k = 0.2
E_p = 1
T = 1000
D = D_0 * np.exp(-E_k / (F.k_B * T))
S = 2.9e-5 * np.exp(-1 / (F.k_B * T))
sample_depth = 1e-3
c_m = (1e5) ** 0.5 * S * 1.0525e5
zeta = (6 * rho_w * np.exp((E_k - E_p) / (F.k_B * T)) + c_m * N_A) / (rho_w * 1e-3)
D_eff = D / (1 + 1 / zeta)
model = F.Simulation()
model.mesh = F.MeshFromVertices(vertices=np.linspace(0, sample_depth, num=100))
model.materials = F.Material(id=1, D_0=D_0, E_D=E_k)
model.T = F.Temperature(value=T)
model.boundary_conditions = [
F.DirichletBC(surfaces=1, value=c_m * N_A, field=0),
F.DirichletBC(surfaces=2, value=0, field=0),
]
trap = F.Trap(
k_0=1.58e7 / N_A,
E_k=E_k,
p_0=1e13,
E_p=E_p,
density=1e-3 * rho_w,
materials=model.materials[0],
)
model.traps = [trap]
model.settings = F.Settings(
absolute_tolerance=1e10, relative_tolerance=1e-10, final_time=100 # s
)
model.dt = F.Stepsize(0.5)
derived_quantities = F.DerivedQuantities([F.HydrogenFlux(surface=2)])
model.exports = [derived_quantities]
model.initialise()
model.run()
Defining initial values
Defining variational problem
Defining source terms
Defining boundary conditions
Time stepping...
Calling FFC just-in-time (JIT) compiler, this may take some time.
Calling FFC just-in-time (JIT) compiler, this may take some time.
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Comparison with exact solution#
Show code cell source
# plot computed solution
t = np.array(derived_quantities.t)
computed_solution = derived_quantities.filter(surfaces=2).data
plt.plot(t, np.abs(computed_solution) / 2, label="FESTIM", linewidth=3)
# plot exact solution
t = np.array(t)
m = np.arange(1, 10001)
# Calculate the exponential part for all m values at once
tau = sample_depth**2 / (np.pi**2 * D_eff)
exp_part = np.exp(-m ** 2 * t[:, None] / tau)
# Calculate the 'add' part for all m values and sum them up for each t
add = 2 * (-1) ** m * exp_part
exact_solution = 1 + add.sum(axis=1) # Sum along the m dimension and add 1 for the initial ones
exact_solution = N_A * exact_solution * c_m * D / (2 * sample_depth)
plt.plot(t, exact_solution, linestyle="--", color="green", label="exact")
plt.xlabel("Time (s)")
plt.ylabel("Downstream flux (H/m2/s)")
plt.legend()
plt.show()
