Diffusion through a semi-infinite slab#
This verification case from TMAP7’s V&V report [3] consists of a semi-infinite slab with no traps under a constant concentration \(C_0\) boundary condition on the left.
FESTIM Code#
Show code cell content
import festim as F
import numpy as np
import sympy as sp
from scipy.special import erf
from matplotlib import pyplot as plt
C_0 = 1 # atom m^-3
D = 1 # m^2 s^-1
profile_time = 25 # s
exact_solution = lambda x, t: C_0 * (1 - erf(x / np.sqrt(4*D*t)))
model = F.Simulation()
### Mesh Settings ###
vertices = np.concatenate([
np.linspace(0, 1, 100),
np.linspace(1, 20, 200),
])
model.mesh = F.MeshFromVertices(vertices)
model.boundary_conditions = [
F.DirichletBC(surfaces=[1], value=C_0, field="solute")
]
model.materials = [F.Material(id=1, D_0=D, E_D=0)]
model.T = F.Temperature(500) # ignored in this problem
model.dt = F.Stepsize(
initial_value=0.01,
stepsize_change_ratio=1.1,
milestones=[profile_time]
)
model.settings = F.Settings(
absolute_tolerance=1e-10,
relative_tolerance=1e-10,
final_time=30
)
test_point_x = 0.45
derived_quantities = F.DerivedQuantities(
[F.PointValue("solute", x=test_point_x)]
)
model.exports = [
derived_quantities,
F.TXTExport(
field="solute",
filename="./tmap_1b_concentration.txt",
times=[profile_time]
),
]
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.
0.0 % 1.0e-02 s Elapsed time so far: 1.5 s
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35.2 % 1.1e+01 s Elapsed time so far: 1.6 s
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57.0 % 1.7e+01 s Elapsed time so far: 1.6 s
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69.0 % 2.1e+01 s Elapsed time so far: 1.6 s
75.9 % 2.3e+01 s Elapsed time so far: 1.6 s
83.3 % 2.5e+01 s Elapsed time so far: 1.6 s
Calling FFC just-in-time (JIT) compiler, this may take some time.
91.5 % 2.7e+01 s Elapsed time so far: 2.4 s
100.0 % 3.0e+01 s Elapsed time so far: 2.4 s
Comparison with exact solution#
The exact solution is given by
\[
c(x, t) = c_0 \left( 1 - \mathrm{erf}\left( \frac{x}{2\sqrt{Dt}} \right) \right)
\]
Show code cell source
# plotting computed data
computed_data = np.genfromtxt("./tmap_1b_concentration.txt", delimiter=",", skip_header=1)
computed_x = computed_data[:, 0]
computed_solution = computed_data[:, 1]
plt.plot(computed_x, computed_solution, label="FESTIM", linewidth = 3)
# plotting exact solution
exact_y = exact_solution(np.array(computed_x), profile_time)
plt.plot(computed_x, exact_y, label="Exact", color="green", linestyle="--")
plt.title(f"Concentration profile at t={profile_time}s")
plt.ylabel("Concentration (atom / m^3)")
plt.xlabel("x (m)")
plt.legend()
plt.show()
Show code cell source
# plotting computed data
computed_solution = derived_quantities[0].data
t = derived_quantities[0].t
plt.plot(t, computed_solution, label="FESTIM", linewidth = 3)
# plotting exact solution
exact_y = exact_solution(test_point_x, np.array(t))
plt.plot(t, exact_y, label="Exact", color="green", linestyle="--")
# plotting TMAP data
tmap_data = np.genfromtxt("./tmap_point_data.txt", delimiter=" ", names=True)
tmap_t = tmap_data["t"]
tmap_solution = tmap_data["tmap"]
plt.scatter(tmap_t, tmap_solution, label="TMAP7", color="purple")
plt.title(f"Concentration profile at x={test_point_x}m")
plt.ylabel("Concentration (atom / m^3)")
plt.xlabel("t (s)")
plt.legend()
plt.show()
