Boundary conditions

The boundary conditions (BCs) are essential to FESTIM simulations. They describe the mathematical problem at the boundaries of the simulated domain. If no BC is set on a boundary, it is assumed that the flux is null. This is also called a symmetry BC.

Basic BCs

These BCs can be used for heat transfer or hydrogen transport simulations.

Imposing the solution

The value of solutions (concentration, temperature) can be imposed on boundaries with festim.DirichletBC:

my_bc = DirichletBC(surfaces=[2, 4], value=10, field=0)

Note

Here, we set field=0 to specify that this BC applies to the mobile hydrogen concentration. 1 would stand for the trap 1 concentration, and "T" for temperature.

The value argument can be space and time dependent by making use of the FESTIM variables x, y, z and t:

from festim import x, y, z, t
my_bc = DirichletBC(surfaces=3, value=10 + x**2 + t, field="T")

To use more complicated mathematical expressions, you can use the sympy package:

from festim import x, y, z, t
import sympy as sp

my_bc = DirichletBC(surfaces=3, value=10*sp.exp(-t), field="T")

• CustomDirichlet

The value of the concentration field can be temperature-dependent (useful when dealing with heat-transfer solvers) with festim.CustomDirichlet:

def value(T):
    return 3*T + 2

my_bc = CustomDirichlet(surfaces=3, function=value, field=0)

Imposing the flux

When the flux needs to be imposed on a boundary, use the festim.FluxBC class.

my_bc = FluxBC(surfaces=3, value=10 + x**2 + t, field="T")

As for the Dirichlet boundary conditions, the flux can be dependent on temperature and mobile hydrogen concentration:

def value(T, mobile):
    return mobile**2 + T

my_bc = CustomFlux(surfaces=3, function=value, field=0)

Hydrogen transport BCs

Some BCs are specific to hydrogen transport. FESTIM provides a handful of convenience classes making things a bit easier for the users.

Recombination flux

A recombination flux can be set on boundaries as follows: \(Kr \, c_\mathrm{m}^n\) (See festim.RecombinationFlux). Where \(Kr\) is the recombination coefficient, \(c_\mathrm{m}\) is the mobile hydrogen concentration and \(n\) is the recombination order.

my_bc = RecombinationFlux(surfaces=3, Kr_0=2, E_Kr=0.1, order=2)

Dissociation flux

Dissociation flux can be set on boundaries as: \(Kd \, P\) (see festim.DissociationFlux). Where \(Kd\) is the dissociation coefficient, \(P\) is the partial pressure of hydrogen.

my_bc = DissociationFlux(surfaces=2, Kd_0=2, E_Kd=0.1, P=1e05)

Kinetic surface model (1D)

Kinetic surface model can be included to account for the evolution of adsorbed hydrogen on a surface with the festim.SurfaceKinetics class. The current class is supported for 1D simulations only. Refer to the Kinetic surface model theory section for more details.

from festim import t
import fenics as f

def k_bs(T, surf_conc, mobile_conc, t):
    return 1e13*f.exp(-0.2/k_b/T)

def k_sb(T, surf_conc, mobile_conc, t):
    return 1e13*f.exp(-1.0/k_b/T)

def J_vs(T, surf_conc, mobile_conc, t):

    J_des = 2e5*surf_conc**2*f.exp(-1.2/k_b/T)
    J_ads = 1e17*(1-surf_conc/1e17)**2*f.conditional(t<10, 1, 0)

    return J_ads - J_des

my_bc = SurfaceKinetics(
    k_bs=k_bs,
    k_sb=k_sb,
    lambda_IS=1.1e-10,
    n_surf=1e17,
    n_IS=6.3e28,
    J_vs=J_vs,
    surfaces=3,
    initial_condition=0,
    t=t
    )

Sievert’s law of solubility

Impose the mobile concentration of hydrogen as \(c_\mathrm{m} = S(T) \sqrt{P}\) where \(S\) is the Sievert’s solubility and \(P\) is the partial pressure of hydrogen (see festim.SievertsBC).

from festim import t

my_bc = SievertsBC(surfaces=3, S_0=2, E_S=0.1, pressure=2 + t)

Henry’s law of solubility

Similarly, the mobile concentration can be set from Henry’s law of solubility \(c_\mathrm{m} = K_H P\) where \(K_H\) is the Henry solubility (see festim.HenrysBC).

from festim import t

my_bc = HenrysBC(surfaces=3, H_0=2, E_H=0.1, pressure=2 + t)

Plasma implantation approximation

Plasma implantation can be approximated by a Dirichlet boundary condition with the class festim.ImplantationDirichlet . Refer to the Theory section for more details.

from festim import t

# instantaneous recombination
my_bc = ImplantationDirichlet(surfaces=3, phi=1e10 + t, R_p=1e-9, D_0=1, E_D=0.1)

# non-instantaneous recombination
my_bc = ImplantationDirichlet(surfaces=3, phi=1e10 + t, R_p=1e-9, D_0=1, E_D=0.1, Kr_0=2, E_Kr=0.2)

# non-instantaneous recombination and dissociation
my_bc = ImplantationDirichlet(surfaces=3, phi=1e10 + t, R_p=1e-9, D_0=1, E_D=0.1, Kr_0=2, E_Kr=0.2, Kd_0=3, E_Kd=0.3, P=4)

Heat transfer BCs

A convective heat flux can be set as \(\mathrm{flux} = - h (T - T_\mathrm{ext})\) (see festim.ConvectiveFlux).

from festim import t

my_bc = ConvectiveFlux(surfaces=3, h_coeff=0.1, T_ext=600 + 10*t)