Subdomains & Materials

Subdomains define different regions within the simulation domain, each assigned specific physical models or materials.

Subdomains are categorized as: 1. Surface subdomains: Regions on the outer boundaries of the simulation domain. 2. Volume subdomains: Regions inside the simulation domain.

Surface Subdomains

Use the festim.SurfaceSubdomain class to define surface subdomains.

from festim import SurfaceSubdomain

my_surface = SurfaceSubdomain(id=1)

The id is a unique identifier for the surface subdomain. It corresponds to mesh tags assigned to the model or can be set during mesh creation using external tools.

Note

If no mesh tags are provided, the surface subdomain ID defaults to 1 on all outer boundaries.

For 1D domains, use the festim.SurfaceSubdomain1D class, which requires an additional x argument to specify the surface position.

from festim import SurfaceSubdomain1D

my_1D_surface = SurfaceSubdomain1D(id=1, x=10)

Custom surface subdomains can be created by subclassing the festim.SurfaceSubdomain class. In this case we can use a custom unit square mesh, and would like to have a defined surface on the top of the domain where y=1.

from dolfinx.mesh import create_unit_square
from mpi4py import MPI
from festim import SurfaceSubdomain

my_mesh = create_unit_square(MPI.COMM_WORLD, 50, 50)

class TopSurface(SurfaceSubdomain):

    # Surface subdomains need a method to locate the facets of the mesh
    def locate_boundary_facet_indices(self, mesh):
        surface_dim = mesh.topology.dim - 1 # dimension of the surface of the domain

        # locate the facets of the mesh where y = 1
        indices = locate_entities(mesh, fdim, lambda x: np.isclose(x[1], 1))
        return indices

Note

The different coordinates x, y, z are represented by x[0], x[1], x[2] in fenics, respectively.

Volume Subdomains

Volume subdomains define distinct regions within the simulation domain and assign materials to these regions.

from festim import VolumeSubdomain, Material

my_material = Material(D_0=1, E_D=1)
my_volume = VolumeSubdomain(id=1, material=my_material)

For 1D domains, use the festim.VolumeSubdomain1D class, which requires a borders argument to specify the domain boundaries where the material is applied.

from festim import VolumeSubdomain1D, Material

my_material = Material(D_0=1, E_D=1)
my_1D_volume = VolumeSubdomain1D(id=1, material=my_material, borders=[0, 1])

Materials

Materials play a key role in hydrogen transport simulations, defining diffusivity, solubility, and thermal properties such as thermal conductivity and heat capacity.

To define a material, use the festim.Material class:

from festim import Material

mat = Material(D_0=2, E_D=0.1)

The festim.Material class requires two arguments:

• D_0: The diffusivity pre-exponential factor (m²/s).

• E_D: The diffusivity activation energy (eV).

Diffusivity is automatically computed using these parameters based on the Arrhenius law.

Additional parameters are required for specific simulations. When considering chemical potential conservation at material interfaces, hydrogen solubility must be specified using:

• name: Name for the material.

• S_0: The solubility pre-exponential factor (units depend on the solubility law: Sievert’s or Henry’s).

• E_S: The solubility activation energy (eV).

• solubility_law: The solubility law, either "festim.SolubilityLaw.HENRY" or festim.SolubilityLaw.SIEVERT.

For transient heat transfer simulations, thermal conductivity, heat capacity, and density must be defined:

• thermal_conductivity: Thermal conductivity (W/m/K).

• heat_capacity: Heat capacity (J/kg/K).

• density: Density (kg/m³).

Temperature-dependent Parameters

Thermal properties can be defined as functions of temperature. For example:

from festim import Material
import ufl

my_mat = Material(
    name="my_fancy_material",
    D_0=2e-7,
    E_D=0.2,
    thermal_conductivity=lambda T: 3 * T + 2 * ufl.exp(-20 * T),
    heat_capacity=lambda T: 4 * T + 8,
    density=lambda T: 7 * T + 5,
)

Integration with HTM

H-transport-materials (HTM) is a Python database of hydrogen transport properties. Using HTM helps prevent copy-paste errors and ensures consistency across simulations by using standardised property values.

HTM can be easily integrated with FESTIM.

Note

This example demonstrates HTM integration with FESTIM v1.4, but the same principle applies to other versions.