Demo: Using the Functions Module

This demo illustrates the usage of the functions module in our library. Specifically, we will demonstrate how to create indicator functions for fictitious domains.

First, we create two domains using the Mesh object. The first mesh, domain represents the entire computational domain, while the second mesh, fictitious, represents a subdomain within the main domain.

\[\Omega_{fictitious} \subset \Omega_{domain}\]

import dolfinx

from flatiron_tk.functions import build_field_scalar_function
from flatiron_tk.functions import build_rank_indicator_function
from flatiron_tk.mesh import RectMesh
from mpi4py import MPI

# Define the domain and fictitious region
domain = RectMesh(0.0, 0.0, 10.0, 10.0, 1/20)
fictitious = RectMesh(2.0, 2.0, 8.0, 8.0, 1/20)

Field Scalar (Indicator) Function

We will then create a function that indicates the presence of the fictitious domain within the main domain. The function will return 1 inside the fictitious domain and 0 outside. Note, the interior/exterior values can be customized.

# Build a scalar function that is 1 inside the fictitious region and 0 outside
inside_value = 1.0
outside_value = 0.0
scalar_function = build_field_scalar_function(domain, fictitious, inside_value, outside_value)

We then save the function to a file for visualization.

# Write the scalar function to an XDMF file
with dolfinx.io.XDMFFile(MPI.COMM_WORLD, 'scalar_function.xdmf', 'w') as xdmf:
    xdmf.write_mesh(domain.msh)
    xdmf.write_function(scalar_function)

Rank Indicator Function

Next, we create a rank indicator function that assigns a unique value to each MPI process. This is useful for visualizing how the domain is partitioned across different processes while debugging parallel computations.

# Build a rank indicator function
rank_function = build_rank_indicator_function(domain)

with dolfinx.io.XDMFFile(MPI.COMM_WORLD, 'rank_indicator_function.xdmf', 'w') as xdmf:
    xdmf.write_mesh(domain.msh)
    xdmf.write_function(rank_indicator_function)

Full Script

import dolfinx

from flatiron_tk.functions import build_field_scalar_function
from flatiron_tk.functions import build_rank_indicator_function
from flatiron_tk.mesh import RectMesh
from mpi4py import MPI

# Define the domain and fictitious region
domain = RectMesh(0.0, 0.0, 10.0, 10.0, 1/20)
fictitious = RectMesh(2.0, 2.0, 8.0, 8.0, 1/20)

# Build a scalar function that is 1 inside the fictitious region and 0 outside
inside_value = 1.0
outside_value = 0.0
scalar_function = build_field_scalar_function(domain, fictitious, inside_value, outside_value)

# Write the scalar function to an XDMF file
with dolfinx.io.XDMFFile(MPI.COMM_WORLD, 'scalar_function.xdmf', 'w') as xdmf:
    xdmf.write_mesh(domain.msh)
    xdmf.write_function(scalar_function)

# This function builds a rank indicator function for the mesh
rank_indicator_function = build_rank_indicator_function(domain)
with dolfinx.io.XDMFFile(MPI.COMM_WORLD, 'rank_indicator_function.xdmf', 'w') as xdmf:
    xdmf.write_mesh(domain.msh)
    xdmf.write_function(rank_indicator_function)