MondaicWe consider a spatial domain (d = 2 or 3), a time interval , and
a diffusion equation of the following form:
with initial conditions
.
Here, denotes the space- and time-dependent diffusive field and are describes external forces.
denotes the first time derivative and the spatial gradient operator.
Furthermore, the scalar parameter and the symmetric second-order diffusion tensor are space-dependent coefficients.
can be related to a Wiener process using the relation
\mathcal{D} = \frac{1}{2} \sigma \sigma\,^T,
which direction-dependent smoothing lengths .
For the special case of and , corresponds to the standard deviation of the Gaussian smoothing in meters.
In the isotropic case, simplifies to a scalar value, in which case we may re-write the diffusion equation as
with and the isotropic smoothing length .
# Standard Python packages
import matplotlib.pyplot as plt
import numpy as np
import os
import toml
# Salvus imports
import salvus.namespace as sn
SALVUS_FLOW_SITE_NAME = os.environ.get("SITE_NAME", "token")mesh = sn.simple_mesh.basic_mesh.CartesianHomogeneousAcoustic2D(
x_max=4.0, y_max=6.0, vp=1000.0, rho=1000.0, max_frequency=5000.0
).create_mesh()
# We don't need any material parameters.
mesh.elemental_fields.clear()
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
<salvus.mesh.data_structures.unstructured_mesh.unstructured_mesh.UnstructuredMesh object at 0x7e0f6837f650>
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
<salvus.mesh.data_structures.unstructured_mesh.unstructured_mesh.UnstructuredMesh object at 0x7e104cd1f750>
sim = sn.simple_config.simulation.Diffusion(mesh=mesh)
sim.domain.polynomial_order = 1
sim.physics.diffusion_equation.time_step_in_seconds = 1e-3
sim.physics.diffusion_equation.courant_number = 0.06
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()sn.api.run(
site_name=SALVUS_FLOW_SITE_NAME,
input_file=sim,
ranks=1,
output_folder="output",
get_all=True,
overwrite=True,
wall_time_in_seconds=600,
)SalvusJob `job_2411201331462771_1364928a3a` running on `local` with 1 rank(s). Site information: * Salvus version: 2024.1.2 * Floating point size: 32
* Downloaded 8.0 MB of results to `output`. * Total run time: 1.41 seconds. * Pure simulation time: 1.05 seconds.
<salvus.flow.executors.salvus_job.SalvusJob at 0x7e0f96cbd110>
mesh = sn.UnstructuredMesh.from_h5(filename="output/out.h5")
mesh
<salvus.mesh.data_structures.unstructured_mesh.unstructured_mesh.UnstructuredMesh object at 0x7e0f5544f010>
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