Resolution with the HDG method

We are here interested in solving the Bestest model with the Hybridized Discontinuous Galerkin method (HDG). Unlike usual methods, this method uses discontinuous elements. Mode more details, see [HDG2020] or [FeelppMath].

1. Equations

Here are the equations of the BESTEST model : they govern the transfer of moisture (equations \(M_\_\)) et heat (equations \(H_\_\)) in the wall and in the room.

\[\begin{align} \left(\rho_m c_m \right)_{\mathrm{eff}} \frac{\partial T_w}{\partial t}+\nabla \cdot\left(-k \nabla T_w - L_{\mathrm{v}} \delta_{\mathrm{p}} \nabla\left(\phi_w p_{\mathrm{sat}}\right)\right) &= 0 \tag{$H_w$}\label{hw}\\ \rho_a c_a \frac{\partial T_r}{\partial t} - h_i \cdot A_w \cdot \left( T_w - T_r \right) - \eta \cdot \rho_a c_a \cdot V_a \cdot \left( T_e - T_r \right) &= H \tag{$H_r$}\label{hr} \end{align}\]
\[\begin{align} \xi\frac{\partial\phi_w}{\partial t} + \nabla\cdot\left(-\xi D_\mathrm{w}\nabla\phi_w-\delta_\mathrm{p}\nabla\left(\phi_wp_\mathrm{sat}\right)\right) = 0 \tag{$M_w$}\label{mw} \\ V_a\frac{\partial w_r}{\partial t} = \eta V_a\left(w_e-w_r\right) + \dot{m}_\mathrm{gen} + h_mA_w\left(w_w-w_r\right) \tag{$M_r$}\label{mr} \end{align}\]

The trategy of resolution, with the HDG method is, at each time step :

  1. Solve the moisture system,

  2. With the moisture from the first point, solve the heat system.

2. Resolution of moisture equations

Equation in the room

To solve the equation \(\ref{mr}\), we use the first order approximation :

\[\frac{\partial w_r}{\partial t} \approx \frac{w_r^{n+1}-w_r^n}{\Delta t}\]

To get the value of \(w_r\) at time \(n+1\), we do this calcul :

\[w_r = \frac{w_r^\text{prev} + \eta V_aw_e + \dot{m}_\text{gen} + h_mA_w w_w^\text{prev}}{\frac{V_a}{\Delta t}+\eta V_a+h_mA_w}\]
on this page, \(X^\text{prev}\) corresponds to the quantity \(X\) at the previous time step of the simulation.

Equation in the wall

The couple the two equation \(\ref{mr}\) and \(\ref{mw}\), we use integral boundary conditions. If \(\q\) corresponds to the moisture flux in the wall, we have :

\[\int_{\Gamma_i} \q\cdot\n = \int_{\Gamma_i} h_{m,i}(w_w^{\text{prev}}-w_r)\]

So we put :

\[I_\text{target} = \int_{\Gamma_i}h_{m,i}(w_r-w_w^{\text{prev}})\]

With the Feel++ toolbox HDG, we can solve the equation \(\ref{mw}\), using the value of \(I_\text{target}\).

maybe there is a problem with the term \(L_v\delta_p\nabla(\phi p_\mathrm{sat})\) which may not be taken in the toolbox
we also have to pay attention : the equation \(\ref{mr}\) deals with \(w\) and \(\ref{mw}\) with \(\phi\)

3. Resolution of heat equations

We do the same steps as before :

  • The first order approximation on the equation \(\ref{hr}\) to get the temperature in the room :

\[T_r = \dfrac{H + \eta\rho_ac_aV_aT_e + h_iA_wT_w + \frac{\rho_ac_a}{\Delta t}T_r^\text{prev}}{\frac{\rho_ac_a}{\Delta t} + h_iA_w + \eta\rho_ac_aV_a}\]

  • The solution of \(\phi_w\) obtained in the previous section is given as a second member of the equation \(\ref{hw}\) solved with the toolbox :

\[\left(\rho_m c_m \right)_{\mathrm{eff}} \frac{\partial T_w}{\partial t}+\nabla \cdot\left(-k \nabla T_w\right) = \nabla\cdot\left( L_{\mathrm{v}} \delta_{\mathrm{p}} \nabla\left(\phi_w p_{\mathrm{sat}}\right)\right)\]

In this situation, the value of \(I_\text{target}\) is \(\int_{\Gamma_i} h_i\left(T_w^\text{prev}-T_r\right)\)

4. Bibliography

  • [] A HDG method for elliptic problems with integral boundary condition: Theory and Applications, Silvia Bertoluzza, Giovanna Guidoboni, Romain Hild, Daniele Prada, Christophe Prud’homme, R. Sacco, Lorenzo Sala, Marcela Szopos, In progess, 2020

  • [] Feel++ Mathematics