Coupled Mixed Poisson
1. Models
The equations of the model are:
\begin{align}
\boldsymbol{u} + k\nabla p &= \boldsymbol{f}\\
\frac{1}{M}\frac{\partial p}{\partial t} + \nabla\cdot \boldsymbol{u} &= g
\end{align}
coupled with a system of ODE:
\begin{align}
\frac{\partial \boldsymbol{y}}{\partial t} + \underline{\underline{A}}(\boldsymbol{y},t)\boldsymbol{y} = s(\boldsymbol{y},t) + b(\boldsymbol{y},t)
\end{align}
where \Pi_1\in \boldsymbol{y} and the boundary condition:
\int_\Gamma \boldsymbol{u}\cdot\boldsymbol{n} = \frac{P_I - \Pi_1}{R_b} \quad \text{ on } \Gamma\\
|\Gamma|p - \int_\Gamma p = 0 \quad \text{ on } \Gamma
with P_I = p_{|\Gamma}.
Most of the configuration is the same as for the Mixed Poisson toolbox, except for the boundary condition where the coupling happens.
2. Boundary Conditions
2.1. Coupling
On the flux/current-density/heat-flux:
Example of a Coupling boundary condition
"Coupling":
{
"buffer":
{
"markers":"top", (1)
"capacitor": "Cbuffer.C", (2)
"resistor": "Rbuffer.R", (3)
"circuit": "$cfgdir/test3d0d_linear/test3d0d_linear.fmu", (4)
"buffer": "Pi_1.phi" (5)
}
}
json
1 | marker for \Gamma |
2 | name of the variable in the FMU for C_b |
3 | name of the variable in the FMU for R_b |
4 | path of the FMU |
5 | name of the variable in the FMU for \Pi_1 |