Simple diffusion with varying temperature

🏷 Tags: 2D MMS steady state

This is an extension of the simple MMS example with a non-homogenous temperature gradient.

We will only consider diffusion of hydrogen in a unit square domain \(\Omega\) at steady state with an homogeneous diffusion coefficient \(D\).

We will assume a temperature gradient of \(T = 300 + x\) over the domain \(\Omega\) so the diffusivity coefficient \(D = D_0 \exp\left[-\frac{E_D}{k_B T}\right]\).

Moreover, a Dirichlet boundary condition will be assumed on the boundaries \(\partial \Omega \).

The problem is therefore:

(8)\[\begin{split} \begin{align} &\nabla \cdot (D(T) \ \nabla{c}) = -S \quad \text{on } \Omega \\ & c = c_0 \quad \text{on } \partial \Omega \end{align} \end{split}\]

The exact solution for mobile concentration is:

(9)\[ \begin{equation} c_\mathrm{exact} = 1 + 2 x^2 + 3 y^2 \end{equation} \]

Injecting (9) in (8), we obtain the expressions of \(S\) and \(c_0\):

\[\begin{split} \begin{align} & S = - 4 D x \frac{E_D}{k_B T^2} - 10 D \\ & c_0 = c_\mathrm{exact} \end{align} \end{split}\]

We can then run a FESTIM model with these values and compare the numerical solution with \(c_\mathrm{exact}\).

FESTIM code

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import festim as F
import matplotlib.pyplot as plt
import numpy as np
import dolfinx
from mpi4py import MPI
import ufl

# Create and mark the mesh
nx = ny = 10
fenics_mesh = dolfinx.mesh.create_unit_square(MPI.COMM_WORLD, nx, ny)

# Create the FESTIM model
my_model = F.HydrogenTransportProblem()

H = F.Species("H")
my_model.species = [H]
my_model.mesh = F.Mesh(fenics_mesh)

D_0 = 2
E_D = 2
material = F.Material(D_0=D_0, E_D=E_D)

volume = F.VolumeSubdomain(id=1, material=material)
boundary = F.SurfaceSubdomain(id=1)

my_model.subdomains = [boundary, volume]

# Variational formulation
exact_solution = lambda x: 1 + 2 * x[0] ** 2 + 3 * x[1] ** 2


T = lambda x: 300 + x[0]


D = lambda x: D_0 * ufl.exp(-E_D / (F.k_B * T(x)))
S = lambda x: -D(x) * (4 * x[0] * E_D / (F.k_B * T(x) ** 2) + 10)

my_model.sources = [
    F.ParticleSource(S, volume=volume, species=H),
]

my_model.boundary_conditions = [
    F.FixedConcentrationBC(subdomain=boundary, value=exact_solution, species=H),
]


my_model.temperature = T

my_model.settings = F.Settings(atol=1e-10, rtol=1e-10, transient=False)

my_model.initialise()
my_model.run()

/home/docs/checkouts/readthedocs.org/user_builds/festim-vv-report/conda/festim-2/lib/python3.11/site-packages/festim/coupled_heat_hydrogen_problem.py:1: TqdmExperimentalWarning: Using `tqdm.autonotebook.tqdm` in notebook mode. Use `tqdm.tqdm` instead to force console mode (e.g. in jupyter console)
  import tqdm.autonotebook

Comparison with exact solution

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def error_L2(u_computed, u_exact, degree_raise=3):
    # Create higher order function space
    degree = u_computed.function_space.ufl_element().degree
    family = u_computed.function_space.ufl_element().family_name
    mesh = u_computed.function_space.mesh
    W = dolfinx.fem.functionspace(mesh, (family, degree + degree_raise))

    # Interpolate exact solution, special handling if exact solution
    # is a ufl expression or a python lambda function
    u_ex_W = dolfinx.fem.Function(W)
    if isinstance(u_exact, ufl.core.expr.Expr):
        u_expr = dolfinx.fem.Expression(u_exact, W.element.interpolation_points)
        u_ex_W.interpolate(u_expr)
    else:
        u_ex_W.interpolate(u_exact)

    # Integrate the error
    error = dolfinx.fem.form(
        ufl.inner(u_computed - u_ex_W, u_computed - u_ex_W) * ufl.dx
    )
    error_local = dolfinx.fem.assemble_scalar(error)
    error_global = mesh.comm.allreduce(error_local, op=MPI.SUM)
    return np.sqrt(error_global)

computed_solution = H.solution

E_l2 = error_L2(computed_solution, exact_solution)

exact_solution_function = dolfinx.fem.Function(computed_solution.function_space)
exact_solution_function.interpolate(exact_solution)
E_max = np.max(np.abs(exact_solution_function.x.array - computed_solution.x.array))

print(f"L2 error: {E_l2:.2e}")
print(f"Max error: {E_max:.2e}")

L2 error: 8.75e-03
Max error: 1.57e-05

import pyvista
from dolfinx.plot import vtk_mesh

pyvista.start_xvfb()
pyvista.set_jupyter_backend("html")


def get_u_grid(computed_solution: dolfinx.fem.Function, label: str):
    u_topology, u_cell_types, u_geometry = vtk_mesh(computed_solution.function_space)
    u_grid = pyvista.UnstructuredGrid(u_topology, u_cell_types, u_geometry)
    u_grid.point_data[label] = computed_solution.x.array.real
    u_grid.set_active_scalars(label)
    return u_grid


u_grid_mobile = get_u_grid(computed_solution, "c_mobile")

u_grid_mobile_exact = get_u_grid(exact_solution_function, "c_mobile_exact")

u_plotter = pyvista.Plotter(shape=(1, 2))
u_plotter.subplot(0, 0)
u_plotter.add_mesh(u_grid_mobile, show_edges=True)
u_plotter.view_xy()

u_plotter.subplot(0, 1)
u_plotter.add_mesh(u_grid_mobile_exact, show_edges=True)
u_plotter.view_xy()


if not pyvista.OFF_SCREEN:
    u_plotter.show()
else:
    figure = u_plotter.screenshot("simple_temperature.png")

/home/docs/checkouts/readthedocs.org/user_builds/festim-vv-report/conda/festim-2/lib/python3.11/site-packages/pyvista/plotting/utilities/xvfb.py:48: PyVistaDeprecationWarning: This function is deprecated and will be removed in future version of PyVista. Use vtk-osmesa instead.
  warnings.warn(

Compute convergence rates

It is also possible to compute how the numerical error decreases as we increase the number of cells. By iteratively refining the mesh, we find that the error exhibits a second order convergence rate. This is expected for this particular problem as first order finite elements are used.

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errors = []
ns = [5, 10, 20, 30, 50, 100, 150]

for n in ns:
    nx = ny = n
    fenics_mesh = fenics_mesh = dolfinx.mesh.create_unit_square(MPI.COMM_WORLD, nx, ny)

    new_model = F.HydrogenTransportProblem()
    new_model.mesh = F.Mesh(fenics_mesh)

    new_model.species = my_model.species
    new_model.subdomains = my_model.subdomains
    new_model.sources = my_model.sources
    new_model.boundary_conditions = my_model.boundary_conditions
    new_model.temperature = my_model.temperature
    new_model.settings = my_model.settings

    new_model.initialise()
    new_model.run()

    computed_solution = H.solution
    errors.append(error_L2(computed_solution, exact_solution))

h = 1 / np.array(ns)

plt.loglog(h, errors, marker="o")
plt.xlabel("Element size")
plt.ylabel("L2 error")

plt.loglog(h, 2 * h**2, linestyle="--", color="black")
plt.annotate(
    "2nd order", (h[0], 2 * h[0] ** 2), textcoords="offset points", xytext=(10, 0)
)

plt.grid(alpha=0.3)
plt.gca().spines[["right", "top"]].set_visible(False)