Boundary conditions

Below we derive the two boundary conditions most commonly used with MoCSI: a Neumann no-heat-flux condition at the bottom and a surface energy balance (SEB) at the top. This is the pairing you will find in most simulations of airless small bodies.

Note

MoCSI’s boundary-condition framework is flexible — additional BC types are available out of the box (e.g. constant heat flux at either boundary, prescribed sinusoidal surface temperature) and more can be added on top of the existing framework. See the boundary condition options in the INI reference for the full list of shipped BC types.

No heat flux

Neumann type boundary condition. Often used to describe the bottom boundary condition of a region where no heat flux into deeper depths can be expected on the modelled time scales. It is a special case of the general prescribed heat flux condition, in which the zero on the left hand side would be replaced by a known source term

\[\frac{\partial T}{\partial x} \bigg\vert_{x = x_{\text{max}}} = 0\]

Surface energy balance equation (SEB)

The standard SEB form reads as

\[Q_{\text{sol.}} + k \, \frac{\partial T}{\partial x} \bigg\vert_{\text{top}} = \sigma \, \varepsilon \, T_{\text{top}}^4 \quad,\]

but within the Finite Element Method framework, the capacitance, and stiffness matrix have to be altered, in order to retain the shape of equation. Further, we have to linearize the Stefan-Boltzmann radiation term in order to retain a linear system of equations, for the cases in which no temperature-dependent physical parameters are used. We linearize around the surface temperature of the last known time step to obtain

\[\begin{split}\sigma \, \varepsilon \, \left(T^{n+1}_{\text{top}}\right)^4 \approx \sigma \, \varepsilon \, \left(T^{n}_{\text{top}}\right)^4 + 4 \, \sigma \, \varepsilon \left(T^{n}_{\text{top}}\right)^3 \, \left(T^{n+1}_{\text{top}} - T^{n}_{\text{top}}\right)\\ \approx S_c + S_{fo} \, T^{n+1}_{\text{top}} \quad,\end{split}\]

with \(S_c = - 3 \, \sigma \, \varepsilon \, \left(T^{n}_{\text{top}}\right)^4\) and \(S_{fo} = 4 \, \sigma \, \varepsilon \, \left(T^{n}_{\text{top}}\right)^3\). The resulting matrix elements read

\[\begin{split}\boldsymbol{C_m} = \frac{c_p \, \rho \, A_{m+1/2} \, l_m}{6} \begin{bmatrix} 0 & 0 \\ 1 & 2 \end{bmatrix} \\ \boldsymbol{K_m} = \frac{k \, A_{m+1/2}}{l_m} \begin{bmatrix} 1 & -1 \\ -1 & 1 \end{bmatrix} + S_{fo} \, A_{m+1/2} \begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix}, \\ \overrightarrow{f} = \frac{A_{m+1/2} \, Q \, l_m}{2} \begin{bmatrix} 0 \\ 1 \end{bmatrix} + \left(S_c + Q_{\text{sol.}}\right) \, A_{m+1/2} \begin{bmatrix} 1 \\ 0 \end{bmatrix} \quad.\end{split}\]

Substituting these matrices into the discretization equation one obtains the SEB equation again for the top temperature in the case of no source or sink terms.