Diffusion equation

We consider a spatial domain ΩRd\Omega \subset \mathbf{R}^d (d = 2 or 3), a time interval I=[0,T]I = [0, T], and a diffusion equation of the following form:

m0(x)tu(x,t)(D(x)u(x,t))=0, m_0(\mathbf{x}) \partial_t u(\mathbf{x},t) - \nabla \cdot \big(\mathcal{D}(\mathbf{x}) \nabla u(\mathbf{x},t)\big) = 0,

with initial conditions

u(x,0)=uinit(x)u(\mathbf{x},0) = u_{\text{init}}(\mathbf{x}).

Here, uu denotes the space- and time-dependent diffusive field and uinitu_{\text{init}} are describes external forces. t\partial_t denotes the first time derivative and \nabla the spatial gradient operator. Furthermore, the scalar parameter m0m_0 and the symmetric second-order diffusion tensor D\mathcal{D} are space-dependent coefficients.

D\mathcal{D} can be related to a Wiener process using the relation

D=12σσT,\mathcal{D} = \frac{1}{2} \sigma \sigma\,^T,

which direction-dependent smoothing lengths σi\sigma_i.

For the special case of m0=1m_0 = 1 and T=1T = 1, σ\sigma corresponds to the standard deviation of the Gaussian smoothing in meters.

In the isotropic case, D\mathcal{D} simplifies to a scalar value, in which case we may re-write the diffusion equation as

m0(x)tu(x,t)(m1(x)u(x,t))=0, m_0(\mathbf{x}) \partial_t u(\mathbf{x},t) - \nabla \cdot \big(m_1(\mathbf{x}) \nabla u(\mathbf{x},t)\big) = 0,

with m1=0.5σ2m_1 = 0.5 * \sigma^2 and the isotropic smoothing length σ\sigma.

Copy
%config Completer.use_jedi = False
# Standard Python packages
import toml
import numpy as np
import matplotlib.pyplot as plt

# Salvus imports
from salvus_mesh.structured_grid_2D import StructuredGrid2D
from salvus_mesh.unstructured_mesh import UnstructuredMesh
import salvus_flow.api
import salvus_flow.simple_config as sc
sg = StructuredGrid2D.rectangle(nelem_x=40, nelem_y=60, max_x=4.0, max_y=6.0)
mesh = sg.get_unstructured_mesh()
mesh.find_side_sets("cartesian")
input_mesh = mesh.copy()
input_mesh.attach_field("some_field", np.random.randn(mesh.npoint))
input_mesh.map_nodal_fields_to_element_nodal()
input_mesh.write_h5("initial_values.h5")
input_mesh
/miniconda/envs/py37/lib/python3.7/site-packages/ipykernel_launcher.py:3: UserWarning: writing mesh with no elemental fields attached
  This is separate from the ipykernel package so we can avoid doing imports until
When run interactively in a Jupyter Notebook, this output cell contains a widget. Click the button below to load a preview of it. Please note that most interactive functionality does not work without a running Python kernel.
<salvus_mesh.unstructured_mesh.UnstructuredMesh at 0x7fafb8bbbeb8>
smoothing_length_in_meters = 0.1

mesh.attach_field("M0", np.ones_like(mesh.get_element_nodes()[:, :, 0]))
mesh.attach_field(
    "M1",
    0.5
    * smoothing_length_in_meters ** 2
    * np.ones_like(mesh.get_element_nodes()[:, :, 0]),
)
mesh.attach_field("fluid", np.ones(mesh.nelem))
mesh
When run interactively in a Jupyter Notebook, this output cell contains a widget. Click the button below to load a preview of it. Please note that most interactive functionality does not work without a running Python kernel.
<salvus_mesh.unstructured_mesh.UnstructuredMesh at 0x7fafb8bbbe80>
sim = sc.simulation.Diffusion(mesh=mesh)

sim.domain.polynomial_order = 1

sim.physics.diffusion_equation.time_step_in_seconds = 1e-3

sim.physics.diffusion_equation.initial_values.filename = "initial_values.h5"
sim.physics.diffusion_equation.initial_values.format = "hdf5"
sim.physics.diffusion_equation.initial_values.field = "some_field"

sim.physics.diffusion_equation.final_values.filename = "out.h5"

sim.output.volume_data.filename = "diffusion.h5"
sim.output.volume_data.format = "hdf5"
sim.output.volume_data.fields = ["phi"]
sim.output.volume_data.sampling_interval_in_time_steps = 10

sim.validate()
salvus_flow.api.run(
    site_name="local",
    input_file=sim,
    ranks=1,
    output_folder="output",
    get_all=True,
    overwrite=True,
)
Job `job_1910230019261334_63de9222bb` running on `local` with 1 rank(s).
Site information:
  * Salvus version: 0.10.6
  * Floating point size: 32
* Downloaded 15.4 MB of results to `output`.
* Total run time: 1.37 seconds.
* Pure simulation time: 0.92 seconds.
<salvus_flow.sites.salvus_job.SalvusJob at 0x7fafb3fcbac8>
mesh = UnstructuredMesh.from_h5(filename="output/out.h5")
mesh
When run interactively in a Jupyter Notebook, this output cell contains a widget. Click the button below to load a preview of it. Please note that most interactive functionality does not work without a running Python kernel.
<salvus_mesh.unstructured_mesh.UnstructuredMesh at 0x7fafb3d82240>
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