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mes_simple_diffusion.py
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mes_simple_diffusion.py
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"""Runs on FESTIM 0.9
"""
import FESTIM as F
import numpy as np
import matplotlib.pyplot as plt
my_model = F.Simulation()
my_model.mesh = F.MeshFromVertices(np.linspace(0, 1.5, num=1000))
my_mat = F.Material(id=1, D_0=2, E_D=0.2)
my_model.materials = F.Materials([my_mat])
my_model.traps = F.Traps(
[F.Trap(k_0=0.01, E_k=0.1, p_0=1, E_p=0.1, materials=my_mat, density=2)]
)
my_model.boundary_conditions = [
F.DirichletBC(surfaces=1, value=2),
F.DirichletBC(surfaces=2, value=0),
]
my_model.T = F.Temperature(300)
my_model.dt = F.Stepsize(1)
my_model.settings = F.Settings(
absolute_tolerance=1e-10, relative_tolerance=1e-10, final_time=2000
)
derived_quantities = F.DerivedQuantities([F.SurfaceFlux("solute", surface=2)])
my_model.exports = F.Exports(
[
derived_quantities,
# F.XDMFExport("solute", checkpoint=False, mode=50),
# F.XDMFExport("1", checkpoint=False, mode=50),
]
)
def exact_flux(t: np.array, model: F.Simulation):
c_0 = model.boundary_conditions[0].value
l = model.mesh.size - model.mesh.start
T = model.T.T(l / 2)
mat = model.materials.materials[0]
D = mat.D_0 * np.exp(-mat.E_D / F.k_B / T)
trap = model.traps.traps[0]
k = trap.k_0 * np.exp(-trap.E_k / F.k_B / T)
p = trap.p_0 * np.exp(-trap.E_p / F.k_B / T)
n = trap.density[0](l / 2)
trap_parameter = p / (k * n) + c_0 / n
# print(trap_parameter)
# print(c_0 / n)
D_eff = D / (1 + 1 / trap_parameter)
m = np.vstack((np.arange(1, 10000),) * t.size)
summation = (-1) ** m * np.exp(-(m**2) * np.pi**2 * D_eff * t[:, None] / l**2)
summation = np.sum(summation, axis=1)
val = c_0 * D / l * (1 + 2 * summation)
return val
def max_flux(model: F.Simulation):
l = model.mesh.size - model.mesh.start
c_0 = model.boundary_conditions[0].value
mat = model.materials.materials[0]
T = model.T.T(l / 2)
D = mat.D_0 * np.exp(-mat.E_D / F.k_B / T)
return c_0 * D / l
my_model.initialise()
my_model.run()
t_final = my_model.settings.final_time
t_2 = np.linspace(0.01, 2000, num=500)
plt.figure(figsize=(6.4, 4))
plt.plot(t_2 / t_final, exact_flux(t_2, my_model) / max_flux(my_model), label="exact")
t = np.array(derived_quantities.data[1:])[:, 0]
flux = -np.array(derived_quantities.data[1:])[:, 1]
flux_ref = exact_flux(t, my_model)
relative_L2_norm = np.linalg.norm(flux - flux_ref, 2) / np.linalg.norm(flux_ref, 2)
print(relative_L2_norm)
plt.plot(t / t_final, flux / max_flux(my_model), label="computed", linestyle="dashed")
plt.ylabel(r"$\varphi / \max{(\varphi)}$")
plt.xlabel("$t/t_f$")
plt.legend()
plt.ylim(0, 1.1)
plt.xlim(0, 1)
plt.gca().spines["right"].set_visible(False)
plt.gca().spines["top"].set_visible(False)
plt.tight_layout()
plt.savefig("mes_festim_effective_diffusion.pdf")
# MAKE DT AND DX VARY
errors = []
dts = np.logspace(1, 3, num=6)
for dt in dts:
my_model.dt = F.Stepsize(dt)
my_model.t = 0
my_model.initialise()
my_model.run()
t_sim = np.array(derived_quantities.data[1:])[:, 0]
flux_ref = exact_flux(t_sim, my_model)
computed_flux = -np.array(derived_quantities.data[1:])[:, 1]
error = np.linalg.norm(computed_flux - flux_ref, 2)
print(error)
errors.append(error)
fig, axs = plt.subplots(1, 2, sharey=True)
plt.sca(axs[0])
plt.plot(dts, errors, marker="+")
plt.xscale("log")
plt.xlabel("$\Delta t$ (s)")
plt.ylabel("L2 error")
plt.gca().spines["right"].set_visible(False)
plt.gca().spines["top"].set_visible(False)
plt.tight_layout()
errors = []
Ns = np.logspace(np.log10(3), 3, num=5)
for N in Ns:
my_model.mesh = F.MeshFromVertices(np.linspace(0, 1.5, num=int(N)))
my_model.dt = F.Stepsize(1)
my_model.t = 0
my_model.initialise()
my_model.run()
t_sim = np.array(derived_quantities.data[1:])[:, 0]
flux_ref = exact_flux(t_sim, my_model)
computed_flux = -np.array(derived_quantities.data[1:])[:, 1]
error = np.linalg.norm(computed_flux - flux_ref, 2)
print(error)
errors.append(error)
hs = [1 / N for N in Ns]
plt.sca(axs[1])
plt.plot(hs, errors, marker="+")
plt.xscale("log")
plt.yscale("log")
plt.xlabel("Element size (m)")
plt.gca().spines["right"].set_visible(False)
plt.gca().spines["top"].set_visible(False)
plt.tight_layout()
plt.savefig("error_vs_element_size_and_dt.png")
plt.savefig("error_vs_element_size_and_dt.pdf")