Simple diffusion case

🏷 Tags: 2D MMS steady state

This is a simple MMS example. We will only consider diffusion of hydrogen in a unit square domain \(\Omega\) at steady state with an homogeneous diffusion coefficient \(D\). Moreover, a Dirichlet boundary condition will be assumed on the boundaries \(\partial \Omega \).

The problem is therefore:

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

The exact solution for mobile concentration is:

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

Injecting (3) in (2), we obtain the expressions of \(S\) and \(c_0\):

(4)\[\begin{align} & S = -10 D \\ & c_0 = c_\mathrm{exact} \end{align}\]

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

FESTIM code

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

# 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 = 2.0
material = F.Material(D_0=D, E_D=0)

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

my_model.subdomains = [boundary, volume]


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

my_model.sources = [
    F.ParticleSource(-10 * D, volume=volume, species=H),
]

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


my_model.temperature = 500

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

First, we compute the \(L^2\)-norm of the error, defined by \(E=\sqrt{\int_\Omega (c-c_\mathrm{exact})^2\mathrm{d} x}\). Secondly, we compute the maximum error at any degree of freedom.

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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: 8.76e-03
Max error: 4.88e-15

The concentration fields can be visualised using pyvista.

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import pyvista
from dolfinx.plot import vtk_mesh

pyvista.start_xvfb()
pyvista.set_jupyter_backend('html')

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["c"] = computed_solution.x.array.real
u_grid.set_active_scalars("c")
u_plotter = pyvista.Plotter()
u_plotter.add_mesh(u_grid, show_edges=False)
contours = u_grid.contour(50)
u_plotter.add_mesh(contours)
u_plotter.view_xy()

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

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u_grid = pyvista.UnstructuredGrid(u_topology, u_cell_types, u_geometry)
u_grid.point_data["c_exact"] = exact_solution_function.x.array.real
u_grid.set_active_scalars("c_exact")
u_plotter = pyvista.Plotter()
u_plotter.add_mesh(u_grid, show_edges=False)
contours = u_grid.contour(50)
u_plotter.add_mesh(contours)
u_plotter.view_xy()
if not pyvista.OFF_SCREEN:
    u_plotter.show()
else:
    figure = u_plotter.screenshot("simple_exact.png")

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.

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)